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8.01 Bernoulli distribution

Worksheet
Bernoulli random variables
1

State whether the following describes a Bernoulli random variable:

a

A die is rolled. X= the number appearing uppermost on the die.

b

Two coins are tossed. X= the number of heads tossed.

c

Two coins are tossed. X=1 if at least one head shows, X=0 if there are no heads showing.

d

A die is rolled. X=1 if an even number shows uppermost, X=0 if an odd number shows uppermost.

2

State whether the following describes a Bernoulli random variable:

a

X=1 if the following spinner lands on the figure 2 if it's spun once, and X = 0 otherwise.

b

X= the numerical result of the figure that the following spinner lands on when it's spun once.

c

X= the numerical result of the figure that the following spinner lands on when it's spun once.

3

A family with three children is chosen at random from the Australian census and whether or not the family has three girls is observed.

a

Define the random variable X that describes the Bernoulli distribution for this situation.

b

Describe the probability of success.

c

Construct the probability distribution table for the random variable X.

4

A die is rolled once in an experiment and the focus is on whether a 4 appears uppermost on the die.

a

Define the random variable X that describes the Bernoulli distribution for this situation.

b

Describe the probability of success.

c

Construct the probability distribution table for the random variable X.

5

A family with four children is chosen at random from the Australian census and whether or not the family has exactly one boy is observed.

a

Define the random variable X that describes the Bernoulli distribution for this situation.

b

Describe the probability of success.

c

Construct the probability distribution table for the random variable X.

6

A burner on a stovetop ignites with a probability of 0.8 on each attempt. Define X as a random variable such that X = 1 if the burner ignites on the first try and X = 0 otherwise.

a

Describe the probability of success.

b

Construct the probability distribution table for the random variable X.

7

A random number generator is programmed to generate the integers 0 to 9 inclusive, but the number 4 is more likely to appear than each of the other integers.

a

Five numbers are generated and the number of 4's are recorded. Can this situation be modelled by a Bernoulli random variable?

b

A single number is generated and whether or not a 4 appears is observed. Can this situation be modelled by a Bernoulli random variable?

c

The probability distribution for the Bernoulli random variable is given in the table. What is the probability of obtaining a 4?

x01
P \left(X = x \right)\dfrac{7}{10}\dfrac{3}{10}
Probability of success
8

The following graphs represent the distribution of a Bernoulli random variable. For each graph, determine the probability of success:

a
b
9

Determine the probability of success if the graph of the probability distribution of a Bernoulli random variable is symmetric.

10

Construct a column graph of a Bernoulli random variable with a probability of success of 0.9.

11

If p is the probability of success, construct a column graph of a Bernoulli random variable where 1 - p = 0.4.

Expected value, variance and standard deviation
12

A die is rolled once in an experiment and the focus is on whether a factor of 6 appears uppermost on the die.

a

Define the random variable X that describes the Bernoulli distribution for this situation.

b

Describe the probability of success.

c

Construct the probability distribution table for the random variable X.

d

Calculate the expected value of X.

e

Calculate the variance of X.

f

Calculate the standard deviation of X.

13

Historically, it rains 15 days in June in Sydney. A random day in June is chosen for an outdoor event in Sydney next year. Define X as a random variable such that X = 1 if it rains on the randomly selected day and X = 0 otherwise.

a

Describe the probability of success.

b

Construct the probability distribution table for the random variable X.

c

Calculate the mean value of the distribution.

d

Calculate the variance of the distribution.

e

Calculate the standard deviation of the distribution.

14

A child has x toy cars and y toy teddy bears in a box, with a total of 10 toys.

a

If the child chooses 6 toys randomly, and the number of toy cars is observed, can this situation be modelled by a Bernoulli random variable?

b

If the child chooses 1 toy randomly, and whether or not it is a toy car is observed, can this situation be modelled by a Bernoulli random variable?

c

The probability distribution for the Bernoulli random variable is given in the table. How many toy cars are there?

x01
P \left(X = x \right)\dfrac{4}{5}\dfrac{1}{5}
d

Calculate the mean of the Bernoulli random variable.

e

Calculate the standard deviation of the Bernoulli random variable.

15

Ten Bernoulli trials are conducted and the expected value is 9.

a

State the probability of success.

b

Calculate the variance of these ten trials.

16

The probability of success of a Bernoulli trial is 0.7. The expected number of successes in n trials is 28.

a

Find the value of n.

b

Calculate the standard deviation of these n trials, correct to two decimal places.

17

The probability of success of a Bernoulli trial X, is 0.75.

a

Calculate the expected value.

b

Calculate the variance.

c

Calculate the standard deviation to two decimal places.

18

The probability of failure of a Bernoulli trial X, is 0.65.

a

Calculate the expected value.

b

Calculate the variance.

c

Calculate the standard deviation to two decimal places.

19

The expected value of the distribution of a Bernoulli random variable X, is 0.3.

a

Calculate the probability of success.

b

Calculate the variance.

c

Calculate the standard deviation to two decimal places.

20

The variance of a Bernoulli random variable X, is \dfrac{6}{25}.

a

Calculate the standard deviation.

b

Calculate the possible values of the expected value, \mu.

21

The variance of a Bernoulli random variable X, is \dfrac{3}{16}.

a

Calculate the standard deviation.

b

Calculate the possible values of the probability of success, p.

22

The standard deviation of a Bernoulli random variable X, is \dfrac{3}{10}.

a

Calculate the variance of X.

b

Calculate the possible values of the probability of success, p.

23

The standard deviation of a Bernoulli random variable X, is \dfrac{2}{5}.

a

Calculate the variance of X.

b

Calculate the expected value E \left( X \right).

24

Find the probability of success, p, of Bernoulli random variable X, if \text{Var}\left(X\right) = 0.4 \times \sigma \left(X\right).

25

A random sample of 10 trials of the same Bernoulli distribution are conducted and the results are tabulated below:

Trial12345678910
Observation0001100101
a

Calculate the mean of the sample.

b

Hence, estimate the probability of success of the distribution.

26

A random sample of 700 trials of the same Bernoulli distribution are conducted and the results are tabulated below:

a

Determine the mean of the sample.

b

Hence, estimate the probability of success of the distribution.

Outcome01
Frequency150550
27

In a random sample of 30 trials of a Bernoulli distribution with a probability of success of 0.2, how many 1's would you expect to see in the sample?

28

In a random sample of 20 trials of a Bernoulli distribution with a probability of success of 0.3, how many 0's would you expect to see in the sample?

29

Consider a random sample of 20 trials of a Bernoulli distribution with a variance of \dfrac{51}{400}.

a

Calculate the possible values of the probability of success p.

b

If p < \dfrac{1}{2}, how many 1's would you expect to see in the sample?

30

Suppose that X is a Bernoulli random variable with a probability of success p. Find the value of p such that \text{Var}\left(X\right) is maximised.

31

Tokens numbered 1 to 20 are placed in a bag, and one is selected at random:

  • Let X = 1 if a prime number is selected, and X = 0 otherwise.

  • Let Y = 1 if a number greater than 8 is selected, and Y = 0 otherwise.

a

Find the probability of success for X.

b

Find the probability of success for Y.

c

Hence, state the mean of X.

d

Hence, state the mean of Y.

e

Calculate the exact standard deviation of X.

f

Calculate the exact standard deviation of Y.

g

Which random variable has the highest standard deviation?

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Outcomes

4.3.1.1

use a Bernoulli random variable as a model for two-outcome situations

4.3.1.2

identify contexts suitable for modelling by Bernoulli random variables

4.3.1.3

recognise and determine the mean 𝑝 and variance 𝑝(1−𝑝) of the Bernoulli distribution with parameter p

4.3.1.4

use Bernoulli random variables and associated probabilities to model data and solve practical problems.

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