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8. Discrete random variables
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VCE 12 Methods 2023

8.07 Applications of binomial distributions

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Worksheet
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Lesson

 

Recall that situations that may be modelled by a binomial distribution are those in which there are a number of independent trials of the same experiment and the individual outcomes are either success or failure. We are interested in the probability that there will be some number $k$k successes among the $n$n trials of the experiment.

 

Modelling with a binomial random variable

Select the brand of calculator you use below to work through an example of using a calculator for calculations concerning the binomial distribution. 

Casio ClassPad

How to use the CASIO Classpad to complete the following tasks involving the binomial distribution.

Robyn does archery every Saturday. The probability that she she hits the bullseye is $0.65$0.65 on each attempt. This Saturday Robyn has shots at the target $18$18 times.

  1. Calculate the probability that she will hit exactly $13$13 bullseyes.

  2. Calculate the probability that she will hit at most $10$10 bullseyes.

  3. Calculate the probability that she hits the bullseye on the first $8$8 attempts and none of the rest.

  4. What is the most likely number of bullseyes she will hit?

  5. What is the expected number of bullseyes she will hit?

  6. Robyn attends target practice for $10$10 consecutive Saturdays. What is the probability that on exactly half of these Saturdays, she hits the bullseye exactly $13$13 times?

  7. One Saturday Robyn decides to continue her target practice until she has hit the bullseye $15$15 times. What is the probability that she needs $24$24 shots at target to achieve her goal?

TI Nspire

How to use the TI Nspire to complete the following tasks involving the binomial distribution. 

Robyn does archery every Saturday. The probability that she she hits the bullseye is $0.65$0.65 on each attempt. This Saturday Robyn has shots at the target $18$18 times.

  1. Calculate the probability that she will hit exactly $13$13 bullseyes.

  2. Calculate the probability that she will hit at most $10$10 bullseyes.

  3. Calculate the probability that she hits the bullseye on the first $8$8 attempts and none of the rest.

  4. What is the most likely number of bullseyes she will hit?

  5. What is the expected number of bullseyes she will hit?

  6. Robyn attends target practice for $10$10 consecutive Saturdays. What is the probability that on exactly half of these Saturdays, she hits the bullseye exactly $13$13 times?

  7. One Saturday Robyn decides to continue her target practice until she has hit the bullseye $15$15 times. What is the probability that she needs $24$24 shots at target to achieve her goal?

 

 

Recall that situations that may be modelled by a binomial distribution are those in which there are a number of independent trials of the same experiment and the individual outcomes are either success or failure. We are interested in the probability that there will be some number $k$k successes among the $n$n trials of the experiment.

 

Modelling with a binomial random variable

Select the brand of calculator you use below to work through an example of using a calculator for calculations concerning the binomial distribution. 

Casio ClassPad

How to use the CASIO Classpad to complete the following tasks involving the binomial distribution.

Robyn does archery every Saturday. The probability that she she hits the bullseye is $0.65$0.65 on each attempt. This Saturday Robyn has shots at the target $18$18 times.

  1. Calculate the probability that she will hit exactly $13$13 bullseyes.

  2. Calculate the probability that she will hit at most $10$10 bullseyes.

  3. Calculate the probability that she hits the bullseye on the first $8$8 attempts and none of the rest.

  4. What is the most likely number of bullseyes she will hit?

  5. What is the expected number of bullseyes she will hit?

  6. Robyn attends target practice for $10$10 consecutive Saturdays. What is the probability that on exactly half of these Saturdays, she hits the bullseye exactly $13$13 times?

  7. One Saturday Robyn decides to continue her target practice until she has hit the bullseye $15$15 times. What is the probability that she needs $24$24 shots at target to achieve her goal?

TI Nspire

How to use the TI Nspire to complete the following tasks involving the binomial distribution. 

Robyn does archery every Saturday. The probability that she she hits the bullseye is $0.65$0.65 on each attempt. This Saturday Robyn has shots at the target $18$18 times.

  1. Calculate the probability that she will hit exactly $13$13 bullseyes.

  2. Calculate the probability that she will hit at most $10$10 bullseyes.

  3. Calculate the probability that she hits the bullseye on the first $8$8 attempts and none of the rest.

  4. What is the most likely number of bullseyes she will hit?

  5. What is the expected number of bullseyes she will hit?

  6. Robyn attends target practice for $10$10 consecutive Saturdays. What is the probability that on exactly half of these Saturdays, she hits the bullseye exactly $13$13 times?

  7. One Saturday Robyn decides to continue her target practice until she has hit the bullseye $15$15 times. What is the probability that she needs $24$24 shots at target to achieve her goal?

 

Practice questions

Question 1

Census data shows that $30%$30% of the population in a particular country have red hair. Find the probability that more than half of a random sample of $6$6 people have red hair.

Give your answer correct to four decimal places.

Question 2

A science exam consisted of $48$48 multiple choice questions, each with $4$4 possible options. Ray guessed the answers to all of the questions at random.

Let $X$X be the number of questions Ray gets correct.

  1. Find $E\left(X\right)$E(X).

  2. Find $Var\left(X\right)$Var(X).

  3. Find the standard deviation $SD\left(X\right)$SD(X).

Question 3

Mohamad is currently applying for graduate job positions. For each application he submits, the probability that it gets short-listed and he gets invited for an interview is $0.02$0.02.

  1. If he applies for $8$8 positions, what is the probability that he will not get a single interview?

    Round your answer to two decimal places.

  2. If he applies for $8$8 positions, what is the probability that he will get at least one interview?

    Round your answer to two decimal places.

  3. If he applies for $n$n positions, what is the probability that he will get at least one interview?

  4. Hence find the minimum number of applications he will have to submit to ensure that the probability that he gets at least one interview is greater than $0.95$0.95.

  5. Mohamad has been applying for $8$8 positions every week over the last $12$12 weeks. Define the random variable $Y$Y, where $Y$Y is the number of weeks Mohamad had at least one interview.

    Using your final answer from part (b), complete the following.

    $Y$Y$\sim$~$B$B$\left(\editable{},\editable{}\right)$(,)
    $P\left(Y=y\right)$P(Y=y) $=$= $\binom{\editable{}}{\editable{}}\times0.15^{\editable{}}\times0.85^{\editable{}}$()×0.15×0.85

     

  6. Calculate the probability that Mohamad had at least one interview per week, for at most $3$3 out of the $12$12 weeks.

    Round your answer to two decimal places.

Question 4

Maria buys a box of $8$8 glasses which is on sale because the box is expected to have around $3$3 broken glasses inside.

  1. Solve for the probability $p$p that a glass is broken.

  2. What is the probability that at most half of the glasses are broken?

    Round your answer to two decimal places.

  3. At the store there are $12$12 boxes in the sale, each containing $8$8 glasses. All of the boxes are expected to have about $3$3 broken glasses. Define the random variable $X$X, where $X$X is the number of boxes containing at most half of the glasses broken.

    Complete the following.

    $P\left(X=x\right)$P(X=x) $=$= $\binom{\editable{}}{\editable{}}\times0.86^{\editable{}}\times0.14^{\editable{}}$()×0.86×0.14

  4. Hence determine the probability that there are exactly $10$10 boxes on display where at most half of the glasses inside are broken.

    Round your answer to two decimal places.

Practice questions

Question 1

Census data shows that $30%$30% of the population in a particular country have red hair. Find the probability that more than half of a random sample of $6$6 people have red hair.

Give your answer correct to four decimal places.

Question 2

A science exam consisted of $48$48 multiple choice questions, each with $4$4 possible options. Ray guessed the answers to all of the questions at random.

Let $X$X be the number of questions Ray gets correct.

  1. Find $E\left(X\right)$E(X).

  2. Find $Var\left(X\right)$Var(X).

  3. Find the standard deviation $SD\left(X\right)$SD(X).

Question 3

Mohamad is currently applying for graduate job positions. For each application he submits, the probability that it gets short-listed and he gets invited for an interview is $0.02$0.02.

  1. If he applies for $8$8 positions, what is the probability that he will not get a single interview?

    Round your answer to two decimal places.

  2. If he applies for $8$8 positions, what is the probability that he will get at least one interview?

    Round your answer to two decimal places.

  3. If he applies for $n$n positions, what is the probability that he will get at least one interview?

  4. Hence find the minimum number of applications he will have to submit to ensure that the probability that he gets at least one interview is greater than $0.95$0.95.

  5. Mohamad has been applying for $8$8 positions every week over the last $12$12 weeks. Define the random variable $Y$Y, where $Y$Y is the number of weeks Mohamad had at least one interview.

    Using your final answer from part (b), complete the following.

    $Y$Y$\sim$~$B$B$\left(\editable{},\editable{}\right)$(,)
    $P\left(Y=y\right)$P(Y=y) $=$= $\binom{\editable{}}{\editable{}}\times0.15^{\editable{}}\times0.85^{\editable{}}$()×0.15×0.85

     

  6. Calculate the probability that Mohamad had at least one interview per week, for at most $3$3 out of the $12$12 weeks.

    Round your answer to two decimal places.

Question 4

Maria buys a box of $8$8 glasses which is on sale because the box is expected to have around $3$3 broken glasses inside.

  1. Solve for the probability $p$p that a glass is broken.

  2. What is the probability that at most half of the glasses are broken?

    Round your answer to two decimal places.

  3. At the store there are $12$12 boxes in the sale, each containing $8$8 glasses. All of the boxes are expected to have about $3$3 broken glasses. Define the random variable $X$X, where $X$X is the number of boxes containing at most half of the glasses broken.

    Complete the following.

    $P\left(X=x\right)$P(X=x) $=$= $\binom{\editable{}}{\editable{}}\times0.86^{\editable{}}\times0.14^{\editable{}}$()×0.86×0.14

  4. Hence determine the probability that there are exactly $10$10 boxes on display where at most half of the glasses inside are broken.

    Round your answer to two decimal places.

Outcomes

U34.AoS4.5

the concepts of a random variable (discrete and continuous), Bernoulli trials and probability distributions, the parameters used to define a distribution and properties of probability distributions and their graphs

U34.AoS4.2

discrete random variables: - specification of probability distributions for discrete random variables using graphs, tables and probability mass functions - calculation and interpretation of mean, 𝜇, variance, 𝜎^2, and standard deviation of a discrete random variable and their use - Bernoulli trials and the binomial distribution, Bi(𝑛, 𝑝), as an example of a probability distribution for a discrete random variable - effect of variation in the value(s) of defining parameters on the graph of a given probability mass function for a discrete random variable - calculation of probabilities for specific values of a random variable and intervals defined in terms of a random variable, including conditional probability

U34.AoS4.6

the conditions under which a Bernoulli trial or a probability distribution may be selected to suitably model various situations

U34.AoS4.11

apply probability distributions to modelling and solving related problems

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