Question

Asked Jun 10, 2019

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A recent poll by the American Automobile Club found that 74.5% of those surveyed are worried about aggressive drivers on the road. If three people are selected at random, what is the probability that (Show how you get these either by formula or calculator)

4.) none will be worried about aggressive drivers on the road?

5.) one will be worried about aggressive drivers on the road?

6.) all will be worried about aggressive drivers on the road?

7.) two will be worried about aggressive drivers on the road?

8.) at least one will be worried about aggressive drivers on the road? 9.) at most two will be worried about aggressive drivers on the road?

Step 1

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Step 2

In this context, a random experiment involving *n* independent trials, such that the outcome of each trial can be classified as either a “success” or a “failure”. The numerical value “1” is assigned to each success and “0” is assigned to each failure.

Moreover, the probability of getting a success in each trial, *p*, remains a constant for all the *n* trials. Denote the probability of failure as *q*. As success and failure are mutually exclusive, *q *= 1 – *p*.

Let the random variable *X* denote the number of successes obtained from the *n* trials. Thus, *X* can take any of the values 0,1,2,…,*3*.

Then, the probability distribution of *X* is a Binomial distribution with parameters (*n*, *p*) and the probability mass function (pmf) of *X*, that is, of a Binomial random variable, is given as:

Step 3

**Parameters of binomial distribution for the given situation:**

The size of the sample taken from the population of people is *n* = 3. It is known that people who is worried and not worried, implying independence among each other. As a result, 3 respondents may be considered as 3 independent trials.

Consider the event that the people who is worried about the aggressive drivers as a “success”. It is given that 74.5% of respondents were worried about aggressive drivers. Thus, the probability that the people who is worried about the aggressive drivers, that...

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