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 “unusual” 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?
The correct answer and explanation is :
a. Probability of Mike guessing correctly
There are 365 days in a non-leap year (ignoring leap years as specified), and Kelly asks Mike to guess the exact day of her birth. The probability that Mike guesses correctly is simply the ratio of one correct guess to the total possible guesses.
$$
P(\text{correct guess}) = \frac{1}{365}
$$
So, the probability that Mike will guess correctly is 1/365, or approximately 0.00274 (about 0.27%).
b. Is it “unusual” for Mike to guess correctly?
To determine if it’s “unusual” for Mike to guess correctly, we can think about what would be considered unusual in terms of probability. A typical threshold for “unusual” events is 5% (0.05) or lower. Since the probability of guessing correctly is 0.00274 (or 0.27%), which is much lower than 5%, we can conclude that it is indeed unusual for Mike to guess correctly.
c. Would you believe Mike’s claim of a lucky guess?
If Mike guessed correctly on the first try, the probability of this happening is so low (about 0.27%) that it would seem highly unlikely to have occurred by pure chance. If you were Kelly, you might question whether Mike already knew her birthdate. While it’s certainly possible that Mike was just lucky, the odds are so low that his claim of a “lucky guess” could seem less credible. You might be more inclined to think he had prior knowledge, especially if the guess was very specific.
d. Probability of a second date if Mike guesses 15 years too high
This part of the problem is more subjective and relies on the dynamic between Kelly and Mike. The probability of them having a second date is influenced by factors such as how Kelly perceives Mike’s behavior, their conversation, chemistry, and how important age is to Kelly.
If Mike guesses Kelly’s age incorrectly by 15 years (too high), it might suggest that Mike either didn’t listen well, doesn’t pay attention to details, or misjudged Kelly’s appearance and personality. This could affect Kelly’s perception of him, but it also depends on how much she values accuracy in such personal details.
There’s no direct mathematical formula for this kind of probability. The decision would largely depend on Kelly’s reaction to the situation, her interest in Mike, and how he handles the incorrect guess. If Mike handles it well, acknowledges the mistake, and they continue to enjoy each other’s company, the probability of a second date might still be high. However, if Kelly is bothered by the mistake, especially if it feels like a significant blunder, she might decide not to have a second date.
In conclusion, this is less about probability and more about social dynamics and Kelly’s personal feelings about the situation.