State whether the following describes a Bernoulli random variable:
A die is rolled. X= the number appearing uppermost on the die.
Two coins are tossed. X= the number of heads tossed.
Two coins are tossed. X=1 if at least one head shows, X=0 if there are no heads showing.
A die is rolled. X=1 if an even number shows uppermost, X=0 if an odd number shows uppermost.
State whether the following describes a Bernoulli random variable:
X=1 if the following spinner lands on the figure 2 if it's spun once, and X = 0 otherwise.
X= the numerical result of the figure that the following spinner lands on when it's spun once.
X= the numerical result of the figure that the following spinner lands on when it's spun once.
A family with three children is chosen at random from the Australian census and whether or not the family has three girls is observed.
Define the random variable X that describes the Bernoulli distribution for this situation.
Describe the probability of success.
Construct the probability distribution table for the random variable X.
A die is rolled once in an experiment and the focus is on whether a 4 appears uppermost on the die.
Define the random variable X that describes the Bernoulli distribution for this situation.
Describe the probability of success.
Construct the probability distribution table for the random variable X.
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.
Define the random variable X that describes the Bernoulli distribution for this situation.
Describe the probability of success.
Construct the probability distribution table for the random variable X.
A burner on a stovetop ignites with a probability of 0.8 on each attempt. X is defined as a random variable such that X = 1 if the burner ignites on the first try and X = 0 otherwise.
Describe the probability of success.
Construct the probability distribution table for the random variable X.
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.
Five numbers are generated and the number of 4's are recorded. Can this situation be modelled by a Bernoulli random variable?
A single number is generated and whether or not a 4 appears is observed. Can this situation be modelled by a Bernoulli random variable?
The probability distribution for the Bernoulli random variable is given in the table.
State the probability of obtaining a 4.
| x | 0 | 1 |
|---|---|---|
| P \left(X = x \right) | \dfrac{7}{10} | \dfrac{3}{10} |
The following graphs represent the distribution of a Bernoulli random variable. For each graph, state the probability of success:
Determine the probability of success if the graph of the probability distribution of a Bernoulli random variable is symmetric.
Construct a column graph of a Bernoulli random variable with a probability of success of 0.9.
If p is the probability of success, construct a column graph of a Bernoulli random variable where 1 - p = 0.4.
A die is rolled once in an experiment and the focus is on whether a factor of 6 appears uppermost on the die.
Define the random variable X that describes the Bernoulli distribution for this situation.
Describe the probability of success.
Construct the probability distribution table for the random variable X.
Calculate the expected value of X.
Calculate the variance of X.
Calculate the standard deviation of X.
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.
Describe the probability of success.
Construct the probability distribution table for the random variable X.
Calculate the mean value of the distribution.
Calculate the variance of the distribution.
Calculate the standard deviation of the distribution.
A child has x toy cars and y toy teddy bears in a box, with a total of 10 toys.
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?
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?
The probability distribution for the Bernoulli random variable is given in the table. How many toy cars are there?
| x | 0 | 1 |
|---|---|---|
| P \left(X = x \right) | \dfrac{4}{5} | \dfrac{1}{5} |
Calculate the mean of the Bernoulli random variable.
Calculate the standard deviation of the Bernoulli random variable.
Ten Bernoulli trials are conducted and the expected value is 9.
State the probability of success.
Calculate the variance of these ten trials.
The probability of success of a Bernoulli trial is 0.7. The expected number of successes in n trials is 28.
Find the value of n.
Calculate the standard deviation of these n trials, correct to two decimal places.
The probability of success of a Bernoulli trial, X, is 0.75.
Calculate the expected value.
Calculate the variance.
Calculate the standard deviation to two decimal places.
The probability of failure of a Bernoulli trial, X, is 0.65.
Calculate the expected value.
Calculate the variance.
Calculate the standard deviation to two decimal places.
The expected value of the distribution of a Bernoulli random variable, X, is 0.3.
Calculate the probability of success.
Calculate the variance.
Calculate the standard deviation to two decimal places.
The variance of a Bernoulli random variable, X, is \dfrac{6}{25}.
Calculate the standard deviation.
Calculate the possible values of the expected value, \mu.
The variance of a Bernoulli random variable, X, is \dfrac{3}{16}.
Calculate the standard deviation.
Calculate the possible values of the probability of success, p.
The standard deviation of a Bernoulli random variable X, is \dfrac{3}{10}.
Calculate the variance of X.
Calculate the possible values of the probability of success, p.
The standard deviation of a Bernoulli random variable, X, is \dfrac{2}{5}.
Calculate the variance of X.
Calculate the expected value E \left( X \right).
Find the probability of success, p, of Bernoulli random variable X, if \text{Var}\left(X\right) = 0.4 \times \sigma \left(X\right).
A random sample of 10 trials of the same Bernoulli distribution are conducted and the results are tabulated below:
| Trial | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|
| Observation | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 1 |
Calculate the mean of the sample.
Hence, estimate the probability of success of the distribution.
A random sample of 700 trials of the same Bernoulli distribution are conducted and the results are tabulated below:
Determine the mean of the sample.
Hence, estimate the probability of success of the distribution.
| Outcome | 0 | 1 |
|---|---|---|
| Frequency | 150 | 550 |
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?
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?
Consider a random sample of 20 trials of a Bernoulli distribution with a variance of \dfrac{51}{400}.
Calculate the possible values of the probability of success p.
If p < \dfrac{1}{2}, how many 1's would you expect to see in the sample?
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.
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.
Find the probability of success for X.
Find the probability of success for Y.
Hence, state the mean of X.
Hence, state the mean of Y.
Calculate the exact standard deviation of X.
Calculate the exact standard deviation of Y.
Which random variable has the highest standard deviation?