In Exercises 25–32, find the probability and answer the questions.. Guessing Birthdays On their first date, Kelly asks Mike to guess the date of her birth, not including the year.

a. What is the probability that Mike will guess correctly? (Ignore leap years.)

b. Would it be unlikely for him to guess correctly on his first try?

c. If you were Kelly, and Mike did guess correctly on his first try, would you believe his claim that he made a lucky guess, or would you be convinced that he already knew when you were born?

d. If Kelly asks Mike to guess her age, and Mike’s guess is too high by 15 years, what is the probability that Mike and Kelly will have a second date?

On their first date, Mike needs to guess Kelly’s birthdate (excluding the year).

Theprobability of an event is the number that shows how much an event is likely to take place.

For an event A, it has the following formula:

$P\left(A\right)=\frac{\text{Number}\text{of}\text{likely}\text{outcomes}}{\text{Total}\text{number}\text{of}\text{outcomes}}$

a.

The total number of possible days in a year = 365.

The number of days favorable to Kelly’s birthday = 1.

Define an event A that mike guessed the birthdate correctly.

Therefore, the probability of guessing the correct birthdate is given as follows:

$$\begin{gathered} P\left(A\right)=\frac{1}{365} \\ =0.00274 \end{gathered}$$

Thus, the probability that Mike guessed the birthdate correctly is 0.00274.

b.

An event is unusual if the probability of occurrence of the event is lesser than 0.05.

As the probability of guessing the correct date on the first try is extremely low, it is quite unlikely for Mike to make the correct guess on the first try.

c.

The event of guessing the correct birthday is unusual (lesser than 0.05).

The probability of taking a random guess that turns out to be the correct birthday is extremely small. It would appear as Mike knew Kelly’s birthday if he were to make a correct guess on his first try.

d.

A wrong guess by 15 years is a big mistake.

If this was a serious mistake, Kelly might think that Mike does not know about her age and reject a second date.

If this was a mistake made in good humor (to tease her), Kelly might not mind and agree to a second date with Mike.