Questions: On their first date, Kelly asks Mike to guess the date of her birth, not including the year. Complete parts a through c below. a. What is the probability that Mike will guess correctly? (Ignore leap years.) (Type an integer or a simplified fraction.) b. Would it be unlikely for him to guess correctly on his first try? A. It is impossible for Mike to guess correctly on his first try, as the probability of a correct guess is 0. B. No, it is not unlikely for Mike to guess correctly on his first try, as the probability of a correct guess is very high. C. Yes, it is unlikely for Mike to guess correctly on his first try, as the probability of a correct guess is very low. D. It is impossible for Mike to not guess correctly on his first try, as the probability of a correct guess is 1. E. Yes, it is unlikely for Mike to guess correctly on his first try, as the probability of a correct guess is very high. F. No, it is not unlikely for Mike to guess correctly on his first try, as the probability of a correct guess is very low. 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? Mike as the probability of a correct guess is

Solution Steps

a. To find the probability that Mike will guess Kelly's birth date correctly, we need to consider the total number of possible dates in a year (excluding leap years). There are 365 days in a non-leap year, so the probability of guessing the correct date is 1 out of 365.

b. To determine if it is unlikely for Mike to guess correctly on his first try, we compare the probability calculated in part (a) to common thresholds for "unlikely" events. A probability of 1/365 is quite low, suggesting it is unlikely.

c. If Mike guesses correctly on his first try, given the low probability, it might be reasonable to suspect he already knew the date rather than it being a lucky guess.

To find the probability that Mike will guess Kelly's birth date correctly, we consider the total number of possible dates in a year, excluding leap years. There are 365 days in a non-leap year. Therefore, the probability of guessing the correct date is:

\[ \frac{1}{365} \approx 0.00274 \]

We compare the probability calculated in Step 1 to a common threshold for "unlikely" events, which is typically 0.05. Since \(0.00274 < 0.05\), it is considered unlikely for Mike to guess correctly on his first try.

Given the low probability of a correct guess, if Mike does guess correctly on his first try, it is reasonable to suspect that he already knew the date rather than it being a lucky guess.

Final Answer