Probability Distributions and Binomial Distributions Review Answer Key

Discrete Random Variables

Binomial Distribution

There are 3 characteristics of a binomial experiment.

1. There are a fixed number of trials. Retrieve of trials every bit repetitions of an experiment. The letter n denotes the number of trials.
2. There are just two possible outcomes, called "success" and "failure," for each trial. The letter p denotes the probability of a success on one trial, and q denotes the probability of a failure on i trial. p + q = 1.
3. The n trials are independent and are repeated using identical conditions. Because the n trials are independent, the event of 1 trial does not help in predicting the outcome of another trial. Another style of proverb this is that for each individual trial, the probability, p, of a success and probability, q, of a failure remain the same. For example, randomly guessing at a true-false statistics question has only ii outcomes. If a success is guessing correctly, and then a failure is guessing incorrectly. Suppose Joe always guesses correctly on any statistics true-false question with probability p = 0.6. Then, q = 0.4. This means that for every truthful-faux statistics question Joe answers, his probability of success (p = 0.6) and his probability of failure (q = 0.4) remain the same.

The outcomes of a binomial experiment fit a binomial probability distribution. The random variable X = the number of successes obtained in the n independent trials.

The mean, μ, and variance, σ ², for the binomial probability distribution are μ = np and σ ² = npq. The standard difference, σ, is then σ = .

Any experiment that has characteristics two and three and where n = 1 is called a Bernoulli Trial (named subsequently Jacob Bernoulli who, in the late 1600s, studied them extensively). A binomial experiment takes identify when the number of successes is counted in one or more Bernoulli Trials.

At ABC College, the withdrawal rate from an elementary physics class is 30% for any given term. This implies that, for any given term, 70% of the students stay in the class for the unabridged term. A "success" could be divers equally an individual who withdrew. The random variable Ten = the number of students who withdraw from the randomly selected elementary physics class.

Try It

The country health lath is concerned nigh the corporeality of fruit available in school lunches. 40-eight percent of schools in the state offering fruit in their lunches every day. This implies that 52% exercise not. What would a "success" be in this case?

Suppose you play a game that you can only either win or lose. The probability that you win any game is 55%, and the probability that y'all lose is 45%. Each game you lot play is independent. If y'all play the game 20 times, write the function that describes the probability that y'all win 15 of the twenty times. Here, if you lot define X as the number of wins, then X takes on the values 0, ane, 2, 3, …, 20. The probability of a success is p = 0.55. The probability of a failure is q = 0.45. The number of trials is n = twenty. The probability question tin be stated mathematically as P(x = 15).

Try It

A trainer is educational activity a dolphin to do tricks. The probability that the dolphin successfully performs the trick is 35%, and the probability that the dolphin does not successfully perform the fob is 65%. Out of twenty attempts, y'all want to observe the probability that the dolphin succeeds 12 times. State the probability question mathematically.

A fair coin is flipped fifteen times. Each flip is independent. What is the probability of getting more than 10 heads? Let X = the number of heads in 15 flips of the fair coin. X takes on the values 0, ane, two, 3, …, 15. Since the coin is fair, p = 0.5 and q = 0.5. The number of trials is n = 15. State the probability question mathematically.

Effort It

A fair, half-dozen-sided die is rolled x times. Each curlicue is independent. You desire to find the probability of rolling a ane more than than three times. State the probability question mathematically.

Approximately 70% of statistics students do their homework in fourth dimension for information technology to be collected and graded. Each pupil does homework independently. In a statistics course of 50 students, what is the probability that at least forty will practise their homework on time? Students are selected randomly.

a. This is a binomial trouble because there is only a success or a __________, in that location are a fixed number of trials, and the probability of a success is 0.70 for each trial.

b. If we are interested in the number of students who do their homework on time, then how do we define X?

b. X = the number of statistics students who do their homework on fourth dimension

c. What values does x take on?

d. What is a "failure," in words?

d. Failure is defined equally a student who does not complete his or her homework on time.

The probability of a success is p = 0.70. The number of trials is n = 50.

e. If p + q = one, then what is q?

f. The words "at least" interpret every bit what kind of inequality for the probability question P(x ____ xl).

f. greater than or equal to (≥)
The probability question is P(ten ≥ xl).

Endeavour It

Sixty-five percent of people laissez passer the state commuter'southward exam on the get-go attempt. A group of l individuals who have taken the driver'southward exam is randomly selected. Requite two reasons why this is a binomial problem.

Annotation for the Binomial: B = Binomial Probability Distribution Function

X ~ B(northward, p)

Read this as "10 is a random variable with a binomial distribution." The parameters are northward and p; north = number of trials, p = probability of a success on each trial.

Information technology has been stated that near 41% of developed workers accept a high school diploma but do not pursue any further education. If 20 developed workers are randomly selected, detect the probability that at most 12 of them have a high school diploma but practice not pursue whatsoever further pedagogy. How many adult workers do you wait to take a loftier school diploma but do not pursue any further pedagogy?

Permit X = the number of workers who accept a high school diploma simply exercise non pursue any further education.

10 takes on the values 0, 1, 2, …, 20 where north = 20, p = 0.41, and q = 1 – 0.41 = 0.59. 10 ~ B(20, 0.41)

Notice P(x ≤ 12). P(ten ≤ 12) = 0.9738. (estimator or reckoner)

Go into ii^nd DISTR. The syntax for the instructions are as follows:

To calculate (ten = value): binompdf(n, p, number) if "number" is left out, the result is the binomial probability tabular array.
To summate P(ten ≤ value): binomcdf(n, p, number) if "number" is left out, the event is the cumulative binomial probability table.
For this problem: After you are in 2^nd DISTR, arrow downwardly to binomcdf. Printing ENTER. Enter xx,0.41,12). The result is P(ten ≤ 12) = 0.9738.

Note

If you want to find P(ten = 12), use the pdf (binompdf). If you lot desire to find P(ten > 12), use one – binomcdf(xx,0.41,12).

The probability that at most 12 workers have a high school diploma but practice non pursue any further teaching is 0.9738.

The graph of 10 ~ B(20, 0.41) is as follows:

The y-axis contains the probability of x, where X = the number of workers who have only a high school diploma.

The number of adult workers that you lot wait to have a high school diploma only non pursue any further education is the mean, μ = np = (twenty)(0.41) = eight.two.

The formula for the variance is σ² = npq. The standard divergence is σ = .
σ = = 2.20.

Try Information technology

Most 32% of students participate in a community volunteer program outside of school. If 30 students are selected at random, observe the probability that at virtually 14 of them participate in a customs volunteer programme outside of schoolhouse. Utilise the TI-83+ or TI-84 figurer to find the reply.

In the 2013 Jerry's Artarama art supplies catalog, at that place are 560 pages. Viii of the pages feature signature artists. Suppose we randomly sample 100 pages. Let X = the number of pages that feature signature artists.

1. What values does x take on?
2. What is the probability distribution? Notice the following probabilities:
   1. the probability that two pages feature signature artists
   2. the probability that at well-nigh vi pages feature signature artists
   3. the probability that more than than three pages feature signature artists.
3. Using the formulas, calculate the (i) mean and (2) standard divergence.

Endeavour It

Co-ordinate to a Gallup poll, 60% of American adults adopt saving over spending. Allow 10 = the number of American adults out of a random sample of 50 who adopt saving to spending.

1. What is the probability distribution for X?
2. Use your figurer to find the following probabilities:
   1. the probability that 25 adults in the sample adopt saving over spending
   2. the probability that at most 20 adults prefer saving
   3. the probability that more than 30 adults prefer saving
3. Using the formulas, summate the (i) mean and (ii) standard divergence of X.

The lifetime risk of developing pancreatic cancer is well-nigh one in 78 (ane.28%). Suppose nosotros randomly sample 200 people. Allow X = the number of people who will develop pancreatic cancer.

1. What is the probability distribution for X?
2. Using the formulas, calculate the (i) mean and (ii) standard deviation of Ten.
3. Use your calculator to detect the probability that at most viii people develop pancreatic cancer
4. Is it more likely that five or half dozen people will develop pancreatic cancer? Justify your answer numerically.

Try Information technology

During the 2013 regular NBA season, DeAndre Jordan of the Los Angeles Clippers had the highest field goal completion charge per unit in the league. DeAndre scored with 61.three% of his shots. Suppose you cull a random sample of 80 shots fabricated by DeAndre during the 2013 season. Allow X = the number of shots that scored points.

1. What is the probability distribution for X?
2. Using the formulas, calculate the (i) mean and (ii) standard divergence of X.
3. Use your calculator to find the probability that DeAndre scored with 60 of these shots.
4. Detect the probability that DeAndre scored with more than than 50 of these shots.

The post-obit example illustrates a problem that is not binomial. It violates the status of independence. ABC College has a student advisory committee fabricated upwards of ten staff members and 6 students. The committee wishes to choose a chairperson and a recorder. What is the probability that the chairperson and recorder are both students? The names of all commission members are put into a box, and two names are drawn without replacement. The offset proper name drawn determines the chairperson and the 2nd proper name the recorder. There are two trials. However, the trials are not independent because the outcome of the first trial affects the issue of the second trial. The probability of a student on the commencement draw is . The probability of a student on the second draw is , when the showtime draw selects a student. The probability is , when the first draw selects a staff member. The probability of cartoon a pupil'southward name changes for each of the trials and, therefore, violates the condition of independence.

Endeavour Information technology

A lacrosse team is selecting a captain. The names of all the seniors are put into a hat, and the first three that are drawn will be the captains. The names are non replaced in one case they are fatigued (ane person cannot be ii captains). You desire to run across if the captains all play the same position. Land whether this is binomial or not and state why.

Chapter Review

A statistical experiment can exist classified equally a binomial experiment if the post-obit atmospheric condition are met:

1. In that location are a stock-still number of trials, n.
2. There are only ii possible outcomes, called "success" and, "failure" for each trial. The letter p denotes the probability of a success on i trial and q denotes the probability of a failure on one trial.
3. The due north trials are contained and are repeated using identical weather.

The outcomes of a binomial experiment fit a binomial probability distribution. The random variable 10 = the number of successes obtained in the n independent trials. The hateful of X tin be calculated using the formula μ = np, and the standard departure is given by the formula σ = .

Formula Review

10 ~ B(n, p) means that the discrete random variable Ten has a binomial probability distribution with n trials and probability of success p.

10 = the number of successes in north independent trials

n = the number of independent trials

X takes on the values x = 0, 1, 2, three, …, due north

p = the probability of a success for whatsoever trial

q = the probability of a failure for any trial

p + q = 1

q = ane – p

The mean of Ten is μ = np. The standard departure of X is σ = .

Use the post-obit information to reply the next viii exercises: The Higher Education Research Constitute at UCLA collected information from 203,967 incoming commencement-time, full-time freshmen from 270 four-year colleges and universities in the U.S. 71.3% of those students replied that, yes, they believe that same-sexual practice couples should have the correct to legal marital status. Suppose that you lot randomly selection eight get-go-time, full-fourth dimension freshmen from the survey. You are interested in the number that believes that same sexual practice-couples should have the right to legal marital status.

In words, define the random variable X.

X = the number that respond "yes"

X ~ _____(_____,_____)

<!– <solution id="id7617599″> B(8,0.713) –>

What values does the random variable 10 take on?

0, 1, 2, 3, 4, 5, 6, vii, eight

Construct the probability distribution part (PDF).

x	P(x)

<!– <solution id="fs-idm163794480″>

–>

On average (μ), how many would you lot look to answer aye?

5.7

What is the standard departure (σ)?

<!– <solution id="id8562245″> 1.2795 –>

What is the probability that at most v of the freshmen reply "yep"?

0.4151

What is the probability that at least 2 of the freshmen reply "yep"?

<!– <solution id="id8562331″> 0.9990 –>

HOMEWORK

Co-ordinate to a recent article the average number of babies born with significant hearing loss (deafness) is approximately two per one,000 babies in a healthy baby nursery. The number climbs to an average of 30 per ane,000 babies in an intensive care nursery.

Suppose that ane,000 babies from healthy babe nurseries were randomly surveyed. Discover the probability that exactly two babies were born deafened.

<!– <solution id="eip-idm92090528″> 0.2709 –>

Utilise the following data to answer the next four exercises. Recently, a nurse commented that when a patient calls the medical advice line claiming to have the influenza, the hazard that he or she truly has the flu (and not simply a nasty cold) is merely almost 4%. Of the next 25 patients calling in claiming to have the flu, we are interested in how many actually take the influenza.

Define the random variable and list its possible values.

X = the number of patients calling in claiming to accept the influenza, who actually have the flu.

Ten = 0, 1, two, …25

State the distribution of X.

<!– <solution id="eip-idm5232336″> B(25,0.04) –>

Observe the probability that at least 4 of the 25 patients actually have the flu.

0.0165

On average, for every 25 patients calling in, how many do yous look to accept the influenza?

<!– <solution id="eip-idm142934512″> one –>

People visiting video rental stores often rent more than ane DVD at a time. The probability distribution for DVD rentals per customer at Video To Go is given (Effigy). There is five-video limit per client at this store, so nobody ever rents more than five DVDs.

ten	P(x)
0	0.03
1	0.fifty
2	0.24
three
four	0.07
five	0.04

1. Describe the random variable X in words.
2. Observe the probability that a customer rents iii DVDs.
3. Observe the probability that a customer rents at least four DVDs.
4. Observe the probability that a customer rents at almost two DVDs.

1. X = the number of DVDs a Video to Go customer rents
2. 0.12
3. 0.xi
4. 0.77

A schoolhouse newspaper reporter decides to randomly survey 12 students to see if they will attend Tet (Vietnamese New Twelvemonth) festivities this year. Based on by years, she knows that eighteen% of students attend Tet festivities. We are interested in the number of students who will attend the festivities.

1. In words, ascertain the random variable Ten.
2. List the values that X may take on.
3. Give the distribution of 10. X ~ _____(_____,_____)
4. How many of the 12 students exercise nosotros look to attend the festivities?
5. Discover the probability that at most four students will attend.
6. Find the probability that more than two students volition attend.

<!– <solution id="id17988422″> X = the number of students who will attend Tet. 0, one, ii, three, four, v, half dozen, 7, 8, nine, x, eleven, 12 X ~ B(12,0.18) 2.16 0.9511 0.3702 –>

Use the post-obit information to answer the next two exercises: The probability that the San Jose Sharks volition win whatever given game is 0.3694 based on a thirteen-year win history of 382 wins out of one,034 games played (every bit of a sure engagement). An upcoming monthly schedule contains 12 games.

The expected number of wins for that upcoming month is:

1. 1.67
2. 12
3. 
4. iv.43

d. 4.43

Allow X = the number of games won in that upcoming month.

What is the probability that the San Jose Sharks win vi games in that upcoming month?

1. 0.1476
2. 0.2336
3. 0.7664
4. 0.8903

<!– <solution id="id17994552″> a –>

What is the probability that the San Jose Sharks win at least five games in that upcoming calendar month

1. 0.3694
2. 0.5266
3. 0.4734
4. 0.2305

c

A student takes a ten-question truthful-false quiz, merely did not study and randomly guesses each answer. Notice the probability that the student passes the quiz with a form of at to the lowest degree 70% of the questions correct.

<!– <solution id="eip-id1171731523478″> X = number of questions answered correctly X ~ B(ten, 0.5) Nosotros are interested in AT To the lowest degree 70% of x questions correct. lxx% of 10 is vii. We want to find the probability that X is greater than or equal to seven. The issue "at least seven" is the complement of "less than or equal to vi". Using your estimator's distribution menu: 1 – binomcdf(10, .5, vi) gives 0.171875 The probability of getting at least lxx% of the ten questions right when randomly guessing is approximately 0.172. –>

A student takes a 32-question multiple-selection test, but did not study and randomly guesses each answer. Each question has three possible choices for the answer. Find the probability that the educatee guesses more than than 75% of the questions correctly.

Six different colored dice are rolled. Of interest is the number of dice that bear witness a ane.

1. In words, define the random variable Ten.
2. List the values that Ten may have on.
3. Give the distribution of 10. Ten ~ _____(_____,_____)
4. On boilerplate, how many dice would you expect to prove a one?
5. Discover the probability that all half dozen dice prove a 1.
6. Is information technology more probable that 3 or that four dice will prove a one? Use numbers to justify your answer numerically.

<!– <solution id="id17531763″> X = the number of dice that prove a i 0, ane, ii, iii, 4, 5, half dozen Ten ~ B ( vi, 1 6 ) 1 0.00002 three dice –>

More than 96 percent of the very largest colleges and universities (more than 15,000 total enrollments) accept some online offerings. Suppose you randomly pick xiii such institutions. We are interested in the number that offer altitude learning courses.

1. In words, define the random variable Ten.
2. List the values that X may take on.
3. Requite the distribution of X. Ten ~ _____(_____,_____)
4. On average, how many schools would you lot await to offering such courses?
5. Find the probability that at well-nigh ten offer such courses.
6. Is it more likely that 12 or that 13 volition offer such courses? Use numbers to justify your answer numerically and answer in a consummate sentence.

1. Ten = the number of college and universities that offering online offerings.
2. 0, ane, 2, …, 13
3. 10 ~ B(13, 0.96)
4. 12.48
5. 0.0135
6. P(x = 12) = 0.3186 P(x = 13) = 0.5882 More than likely to go thirteen.

Suppose that almost 85% of graduating students attend their graduation. A group of 22 graduating students is randomly chosen.

1. In words, ascertain the random variable X.
2. List the values that Ten may take on.
3. Give the distribution of X. X ~ _____(_____,_____)
4. How many are expected to attend their graduation?
5. Find the probability that 17 or 18 attend.
6. Based on numerical values, would you lot be surprised if all 22 attended graduation? Justify your answer numerically.

<!– <solution id="fs-idm62699008″> 10 = the number of students who attend their graduation 0, 1, 2, …, 22 Ten ~ B(22, 0.85) 18.vii 0.3249 P(x = 22) = 0.0280 (less than 3%) which is unusual –>

At The Fencing Heart, lx% of the fencers apply the foil as their master weapon. We randomly survey 25 fencers at The Fencing Center. We are interested in the number of fencers who do not apply the foil as their principal weapon.

1. In words, define the random variable X.
2. List the values that X may take on.
3. Give the distribution of X. X ~ _____(_____,_____)
4. How many are expected to non to use the foil equally their main weapon?
5. Find the probability that six do not use the foil as their primary weapon.
6. Based on numerical values, would yous be surprised if all 25 did non employ foil as their main weapon? Justify your reply numerically.

1. X = the number of fencers who do not use the foil as their main weapon
2. 0, 1, 2, iii,… 25
3. X ~ B(25,0.40)
4. 10
5. 0.0442
6. The probability that all 25 not use the foil is almost zero. Therefore, it would be very surprising.

Approximately 8% of students at a local loftier school participate in afterwards-school sports all four years of high school. A group of 60 seniors is randomly chosen. Of interest is the number who participated in after-schoolhouse sports all four years of loftier schoolhouse.

1. In words, define the random variable Ten.
2. List the values that X may take on.
3. Requite the distribution of X. 10 ~ _____(_____,_____)
4. How many seniors are expected to take participated in after-school sports all four years of high schoolhouse?
5. Based on numerical values, would you be surprised if none of the seniors participated in after-school sports all iv years of high school? Justify your respond numerically.
6. Based upon numerical values, is it more probable that iv or that five of the seniors participated in after-schoolhouse sports all four years of high school? Justify your reply numerically.

<!– <solution id="fs-idm58938512″> X = the number of high school students who participate in later school sports all four years of high schoolhouse. 0, i, ii, …, lx X ~ B(threescore, 0.08) 4.8 Yes, P(x = 0) = 0.0067, which is a small-scale probability P(x = 4) = 0.1873, P(x = v) = 0.1824. More probable to go four. –>

The chance of an IRS inspect for a revenue enhancement return with over ?25,000 in income is most ii% per yr. We are interested in the expected number of audits a person with that income has in a 20-year menstruation. Assume each year is independent.

1. In words, define the random variable X.
2. List the values that X may take on.
3. Give the distribution of Ten. X ~ _____(_____,_____)
4. How many audits are expected in a 20-yr catamenia?
5. Find the probability that a person is not audited at all.
6. Observe the probability that a person is audited more than twice.

1. X = the number of audits in a xx-year catamenia
2. 0, 1, 2, …, 20
3. X ~ B(20, 0.02)
4. 0.4
5. 0.6676
6. 0.0071

It has been estimated that simply nigh 30% of California residents have acceptable earthquake supplies. Suppose you randomly survey 11 California residents. We are interested in the number who have adequate earthquake supplies.

1. In words, ascertain the random variable X.
2. Listing the values that X may take on.
3. Requite the distribution of X. X ~ _____(_____,_____)
4. What is the probability that at least 8 take adequate earthquake supplies?
5. Is it more than likely that none or that all of the residents surveyed volition have acceptable earthquake supplies? Why?
6. How many residents exercise you wait will have adequate convulsion supplies?

<!– <solution id="id16028424″> 10 = the number of California residents who practise have adequate earthquake supplies. 0, 1, ii, 3, 4, 5, 6, 7, eight, 9, 10, 11 B(11, 0.30) 0.0043 P(10 = 0) = 0.0198. P(x = xi) = 0 or none 3.3 –>

There are 2 like games played for Chinese New year's day and Vietnamese New Yr. In the Chinese version, off-white dice with numbers 1, 2, 3, four, v, and 6 are used, along with a board with those numbers. In the Vietnamese version, off-white dice with pictures of a gourd, fish, rooster, crab, crayfish, and deer are used. The board has those six objects on it, also. We will play with bets being ?1. The player places a bet on a number or object. The "house" rolls 3 dice. If none of the dice show the number or object that was bet, the house keeps the ?one bet. If one of the dice shows the number or object bet (and the other two do not prove information technology), the player gets back his or her ?1 bet, plus ?1 profit. If 2 of the dice testify the number or object bet (and the third die does non evidence it), the actor gets dorsum his or her ?1 bet, plus ?two profit. If all three die show the number or object bet, the histrion gets back his or her ?1 bet, plus ?3 profit. Let X = number of matches and Y = profit per game.

1. In words, define the random variable X.
2. Listing the values that X may take on.
3. Requite the distribution of X. X ~ _____(_____,_____)
4. Listing the values that Y may take on. And then, construct one PDF tabular array that includes both 10 and Y and their probabilities.
5. Calculate the boilerplate expected matches over the long run of playing this game for the role player.
6. Calculate the average expected earnings over the long run of playing this game for the player.
7. Determine who has the advantage, the role player or the house.

According to The World Bank, only 9% of the population of Uganda had access to electricity as of 2009. Suppose we randomly sample 150 people in Uganda. Let X = the number of people who have admission to electricity.

1. What is the probability distribution for 10?
2. Using the formulas, summate the mean and standard departure of X.
3. Use your calculator to notice the probability that 15 people in the sample have access to electricity.
4. Find the probability that at most ten people in the sample have admission to electricity.
5. Find the probability that more than 25 people in the sample have access to electricity.

<!– <solution id="fs-idm37905376″> X ~ B(150,0.09) Mean = np = 150(0.09) = 13.five Standard Deviation = npq =   150(0.09)(0.91) ≈ 3.5050 P(10 = 15) = binompdf(150, 0.09, xv) = 0.0988 P(x ≤ 10) = binomcdf(150, 0.09, 10) = 0.1987 P(x > 25) = 1 – P(x ≤ 25) = ane – binomcdf(150, 0.09, 25) = ane – 0.9991 = 0.0009 –>

The literacy rate for a nation measures the proportion of people age 15 and over that tin read and write. The literacy rate in Afghanistan is 28.one%. Suppose you choose 15 people in Afghanistan at random. Allow X = the number of people who are literate.

1. Sketch a graph of the probability distribution of X.
2. Using the formulas, summate the (i) hateful and (2) standard difference of X.
3. Discover the probability that more than than v people in the sample are literate. Is it is more likely that three people or 4 people are literate.

Glossary

Binomial Experiment

a statistical experiment that satisfies the post-obit three conditions:

1. There are a fixed number of trials, n.
2. At that place are only ii possible outcomes, called "success" and, "failure," for each trial. The letter p denotes the probability of a success on one trial, and q denotes the probability of a failure on ane trial.
3. The n trials are contained and are repeated using identical conditions.

Bernoulli Trials

an experiment with the post-obit characteristics:

1. There are simply 2 possible outcomes called "success" and "failure" for each trial.
2. The probability p of a success is the same for any trial (so the probability q = i − p of a failure is the same for any trial).

Binomial Probability Distribution

a discrete random variable (RV) that arises from Bernoulli trials; there are a stock-still number, due north, of contained trials. "Independent" means that the result of any trial (for instance, trial 1) does not affect the results of the post-obit trials, and all trials are conducted under the same conditions. Nether these circumstances the binomial RV X is defined as the number of successes in due north trials. The note is: X ~ B(northward, p). The hateful is μ = np and the standard departure is σ = . The probability of exactly x successes in n trials is
P(Ten = ten) = p ^x q ^{n − x}.

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Source: https://opentextbc.ca/introstatopenstax/chapter/binomial-distribution/

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