An individual repeatedly attempts to pass a driving test. Suppose that the probability of passing the test with each attempt is 0.25, and that the results of successive tests are independent. Let X be the number of tests taken until the individual passes (a) Find the probability mass function of X (b) Evaluate the probability of passing the test with three or less attempts. (c) Evaluate the probability of passing the test with five or more attempts.

a) Our random variable X="number of tests taken until the individual passes" follows a geomteric distribution with probability of success p=0.25

For this case the probability mass function would be given by:

b)

And adding the values we got:

c)

And we can find the individual probabilities:

Step-by-step explanation:

Previous concepts

The geometric distribution represents "the number of failures before you get a success in a series of Bernoulli trials. This discrete probability distribution is represented by the probability density function:"

Let X the random variable that measures the number of trials until the first success, we know that X follows this distribution:

Part a

Our random variable X="number of tests taken until the individual passes" follows a geomteric distribution with probability of success p=0.25

For this case the probability mass function would be given by:

Part b

We want this probability:

We find the individual probabilities like this:

And adding the values we got:

Part c

For this case we want this probability:

And we can use the complement rule like this:

And we can find the individual probabilities:

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