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8.02 Binomial distribution

Worksheet
Binomial random variables
1

In a binomial experiment:

a

Are the trials independent or dependent?

b

How many possible outcomes are there for each trial?

2

State whether following scenarios describe a binomial random variable:

a

Rolling a six-sided die 20 times and counting the number of times the die lands on 1.

b

The sum of the outcomes of rolling a six-sided die 10 times.

c

Selecting 70 random people from the population and recording the number of females.

d

A particular coin is more likely to land on heads than tails. Tossing this coin 10 times and recording the number of times it lands on heads.

e

Drawing a marble with replacement from a bag containing purple, green and yellow marbles, and noting the number of purple marbles.

f

Drawing a marble with replacement from a bag containing purple, green and yellow marbles, and noting the number of red marbles.

3

Consider the binomial probability formula:P \left( x \right) = {}^{n}C_{x} p^{x} \left(1 - p\right)^{n - x}where P \left( x \right) is the probability of obtaining x successes in n independent trials.

a

What does p represent in the formula?

b

What does \left(1 - p\right) represent in the formula?

4

For a binomial probability distribution with n, trials and probability of success p, Ned performed the following calculation:\binom{n}{2} p^{2} \times 0.7^{4}

a

State the probability of success.

b

State the number of trials.

c

What probability is Ned trying to find with this calculation?

5

Consider the binomial power \left(p + q\right)^{4}.

a

Expand \left(p + q\right)^{4}.

b

We can interpret the terms in the expansion in terms of probabilities in a binomial distribution, with p the probability of success and q the probability of failure. What probability would the term \binom{4}{2} p^{2} q^{2} in the expansion represent?

Distributions of binomial random variables
6

Consider the three graphs below that show binomial distributions for n = 10 and p = 0.15, 0.5 and 0.85:

a

Which value of p corresponds to Graph A?

b

Which value of p corresponds to Graph B?

c

Which value of p corresponds to Graph C?

7

Consider a binomial distribution of a random variable X with p = 0.5 and n = 17.

a

Is P \left( \text{X = 5} \right) greater than, less than or equal to P \left( \text{X = 12} \right)?

b

Is the shape of the graph of this binomial distribution symmetric, positively skewed or negatively skewed?

8

Consider a binomial distribution of a random variable X with p = 0.13 and n = 11.

a

Is P \left( \text{X = 4} \right) greater than, less than or equal to P \left( \text{X = 7} \right)?

b

Is the shape of the graph of this binomial distribution symmetric, positively skewed or negatively skewed?

9

Consider a binomial distribution of a random variable X with p = 0.83 and n = 17.

a

Is P \left( \text{X = 2} \right) greater than, less than or equal to P \left( \text{X = 15} \right)?

b

Is the shape of the graph of this binomial distribution symmetric, positively skewed or negatively skewed?

10

For each of the followng table of values for the binomial distribution X:

i

Describe the shape of the graph as positively skewed, symmetric, or negatively skewed.

ii

Calculate the value of p correct to two decimal places.

a
x012345678
P\left(X=x\right)0.0040.0310.1090.2190.2740.2190.1090.0310.004
b
x012345678
P\left(X=x\right)0.1520.3230.30.160.0530.0110.00100
c
x012345678
P\left(X=x\right)000.0010.0090.0460.1470.2940.3360.168
Probability for binomial random variables
11

Find the value of the following:

a
{}^{5}C_{4} \times \left(0.1\right)^{4} \times 0.9 + {}^{5}C_{5} \times \left(0.1\right)^{5} \times \left(0.9\right)^{0}
b
{}^{5}C_{3} \times \left(0.8\right)^{3} \times \left(0.2\right)^{2} + {}^{5}C_{4} \times \left(0.8\right)^{4} \times 0.2 + {}^{5}C_{5} \times \left(0.8\right)^{5} \times \left(0.2\right)^{0}
12

A binomial random variable X has 13 independent trials and a probability of success of 0.4.

a

Using the definition for the binomial random variable X, write the formula for P \left( X = 2 \right).

b

Calculate P \left( X = 2 \right) to four decimal places.

13

For a binomial random variable X, P \left(X=3\right) = {}^{10}C_{3} \times 0.6^{3} \times 0.4^{7}.

a

Calculate P \left( X = 3 \right) to four decimal places.

b

State the number of trials in this experiment.

c

State the probability of success in this experiment.

14

X is a binomial variable with the probability mass function:P(X = k)={}^{4}C_{k} \times \left(0.4\right)^{k} \times \left(0.6\right)^{4 - k} \text{ for }k = 0, 1, 2, 3, 4

a

State the number of trials for this distribution.

b

State the probability of success.

c

Wrie an expression for P \left(X = 2 \right).

d

How many ways can we get 2 successes in the 4 trials?

e

Calculate the probability P \left(X = 2 \right).

15

X is a binomial variable with the probability mass function:P(X = k)={}^{3}C_{k} \times \left(0.3\right)^{k} \times \left(0.7\right)^{3 - k} \text{ for } k = 0, 1, 2, 3

Complete the table:

x0123
P \left( X = x \right)0.3430.441
16

Consider the random variable X, which has distribution X \sim B\left(4, 0.1 \right).

a

Write an expression for P \left(X = 1 \right).

b

Write an expression for P \left(X = 2 \right).

c

Complete the following table:

x01234
P \left( X = x \right)0.65610.0001
17

A binomial random variable X is defined as:P \left(X=x\right) = {}^{20}C_{x} \times 0.75^{x} \times 0.25^{20 - x} \text{ for }x = 1, 2, \ldots, 20

a

State the number of trials in this experiment.

b

State the probability of success in this experiment.

c

Calculate P \left( X = 11 \right) to four decimal places.

18

X is a binomial variable with the probability mass function:P(X = k)={}^{6}C_{k} \times \left(0.8\right)^{k} \times \left(0.2\right)^{6 - k} \text{ for } k = 0, 1, 2, 3, 4, 5, 6

a

State the number of trials for this distribution.

b

What is the probability of failure?

c

Calculate the probability P \left(X = 3 \right).

d

Calculate the probability P \left(X \leq 3 \right).

19

X is a binomial variable with the probability mass function:P(X = k)={}^{5}C_{k} \times \left(0.4\right)^{k} \times \left(0.6\right)^{5 - k} \text{ for } k = 0, 1, 2, 3, 4, 5

a

Calculate P(X \geq 3).

b

Calculate P(X < 3).

20

X is a binomial variable with the probability mass function:P(X = k)={}^{5}C_{k} \times \left(0.2\right)^{k} \times \left(0.8\right)^{5 - k} \text{ for } k = 0, 1, 2, 3, 4, 5

a

State the number of trials for this distribution.

b

State the probability of success.

c

Calculate the probability P \left(X > 3 \right).

d

What is the most likely number of successes?

e

Calculate the mean of the distribution.

f

Calculate the variance of the distribution.

21

Consider the random variable X, which has distribution X \sim B\left(8, \dfrac{4}{5} \right).

a

State the number of trials for this distribution.

b

State the probability of success.

c

Calculate the probability P \left(X > 4 \right) to two decimal places.

d

What is the most likely number of successes?

e

Calculate the mean of the distribution.

f

Calculate the variance of the distribution.

22

X is a binomial variable with a probability distribution given in the table:

x012345
P ( X = x )0.002430.028350.13230.30870.360150.16807
a

Find P(X \geq 4).

b

Find P(X < 3).

23

The following table represents a probability distribution of a binomial random variable:

x01234
P \left( X = x \right)0.24010.41160.26460.07560.0081
a

State the number of trials for this distribution.

b

Calculate the probability P \left(X = 3 \right).

c

Calculate the probability P \left( X < 3 \right).

d

Using P \left(X = 4 \right), calculate the probability of success p.

e

Calculate E \left( X \right).

f

Calculate the standard deviation of the distribution.

24

The table represents a cumulative probability distribution of a binomial random variable:

x012345
P \left( X \leq x \right)0.010240.087040.317440.663040.922241
a

State the number of trials for this distribution.

b

Calculate P \left(X \leq 2 \right).

c

Calculate P \left( X \geq 3 \right).

d

Calculate P \left( X = 5 \right).

e

Using P \left(X = 5 \right), calculate the probability of success p.

f

Calculate the standard deviation of the distribution.

25

The following graph represents a probability distribution of a binomial random variable:

a

State the number of trials for this distribution.

b

State P \left(X = 3 \right).

c

Calculate P \left( X < 3 \right).

d

Using P \left(X = 4 \right), calculate the probability of success p.

e

Calculate E \left( X \right).

f

Calculate the standard deviation of the distribution.

26

The following graph represents a cumulative probability distribution of a binomial random variable:

a

State the number of trials for this distribution.

b

Calculate P \left(X \leq 2 \right).

c

Calculate P \left( X \geq 3 \right).

d

Calculate P \left( X = 5 \right).

e

Using P \left(X = 5 \right), calculate the probability of success p.

f

Calculate the standard deviation of the distribution.

27

Calculate the probability of the following, rounding your answers to two decimal places:

a

Getting exactly 18 successes in 30 independent trials if the probability of success in each trial is 0.7.

b

Getting at least 10 successes in 80 independent trials if the probability of success in each trial is p = 0.1.

c

Getting at most 22 successes in 25 independent trials if the probability of success in each trial is 0.85.

d

Getting more than 15 successes in 33 independent trials if the probability of success in each trial is 0.45.

e

Getting less than 12 successes in 50 independent trials if the probability of success in each trial is 0.28.

28

P \left(X = 2 \right) = 0.3456 for a binomial random variable X with 4 trials.

a

If success is more likely, find the probability of success p.

b

Calculate E \left(X\right).

c

Calculate \text{Var} \left(X\right).

29

E \left( X \right) = 6 for a binomial random variable X with 10 trials.

a

Find the probability of success p.

b

Calculate \text{Var} \left(X\right).

30

The expected value of a binomial distribution is 8 and the variance is 6.4.

a

Find the number of trials, n.

b

Find the probability of success, p.

31

E \left( X \right) = \dfrac{15}{2} for a binomial random variable X with probability of success \dfrac{3}{10}.

a

Find the number of trials n.

b

Calculate the standard deviation of X.

32

E \left(X\right) = 2 and \text{Var} \left(X\right) = 1.6 for a binomial random variable X.

a

Find the probability of success p.

b

Find the number of trials n.

33

E \left(X\right) = 4.8 and \sigma \left(X\right) = \sqrt{2.88} for a binomial random variable X.

a

Find the probability of success p.

b

Find the number of trials n.

Applications
34

A certain disease has a survival rate of 64\%. Of the next 110 people who contract the disease, how many would you expect to survive? Round your answer to the nearest whole number.

35

The probability of a particular tennis player getting their first serve in is 0.72. If she serves 90 times in a match, how many times would she expect to get her first serve in? Round your answer rounded to the nearest whole number.

36

A fair die is rolled 5 times and the number of fours noted.

a

On each trial, what is the probability of success?

b

Calculate the probability of rolling exactly 2 fours, correct to four decimal places.

c

Calculate the probability of rolling exactly 3 fours, correct to four decimal places.

37

David owns 3 yellow mugs and 8 blue mugs. Each morning, he chooses a mug at random for his cup of tea before work, 5 days a week. The mug is washed and returned before the next day.

a

Each day, what is the probability of choosing a yellow mug?

b

What is the probability that he used a yellow mug exactly twice this past work week? Round your answer to four decimal places.

c

What is the probability that he never used a yellow mug this past work week? Round your answer to four decimal places.

38

At a local Italian restaurant, 16 people are in line to order. The probability that any one person will choose pizza is 23\%.

a

What is the probability that exactly 4 people choose pizza? Round your answer to four decimal places.

b

What is the probability that exactly half of the people choose pizza? Round your answer to four decimal places.

39

Consider the binomial \left(\dfrac{1}{5} + \dfrac{4}{5}\right)^{5}.

a

Write the formula for the term of the expansion that contains \left(\dfrac{4}{5}\right)^{3}.

b

A box of chocolates is \dfrac{1}{5} dark chocolate and \dfrac{4}{5} milk chocolate. If 5 chocolates are selected at random with replacement, describe what the term from part (a) would tell us.

c

Calculate the probability of selecting exactly 1 dark chocolate.

d

Calculate the probability of selecting at most 1 dark chocolate.

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Outcomes

4.3.2.1

understand the concepts of Bernoulli trials and the concept of a binomial random variable as the number of β€˜successes’ in 𝑛 independent Bernoulli trials, with the same probability of success 𝑝 in each trial

4.3.2.2

identify contexts suitable for modelling by binomial random variables

4.3.2.3

determine and use the probabilities 𝑃(𝑋=π‘Ÿ) = {nr} 𝑝^π‘Ÿ(1 βˆ’ 𝑝)^(n-r) associated with the binomial distribution with parameters 𝑛 and p

4.3.2.4

calculate the mean 𝑛p and variance 𝑛p(1βˆ’π‘) of a binomial distribution using technology and algebraic methods

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