Question

Two friends are born in the year 2000. The probability that they have the same birthday

• A. $\frac{1}{365}$
• B. $\frac{364}{365}$
• C. $\frac{365}{366}$
• D. $\frac{1}{366}$

Answer 1

The Correct Option is D

Step 1: Analyze the problem.

The year 2000 is a leap year, so it has 366 days. Each friend can be born on any of these 366 days, and we assume that all days are equally likely.

Step 2: Calculate the total number of possible outcomes.

The first friend can be born on any of the 366 days, and the second friend can also be born on any of the 366 days. Thus, the total number of possible outcomes is:

$$366 \times 366.$$

Step 3: Calculate the favorable outcomes.

For the two friends to have the same birthday, they must both be born on the same day. There are 366 possible days on which this can happen (one for each day of the year). Thus, the number of favorable outcomes is:

$$366.$$

Step 4: Compute the probability.

The probability is the ratio of favorable outcomes to total outcomes:

$$\text{Probability} = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{366}{366 \times 366}.$$

Simplify:

$$\text{Probability} = \frac{1}{366}.$$

Final Answer: The probability that the two friends have the same birthday is $\mathbf{\frac{1}{366}}$, which corresponds to option $\mathbf{(4)}$.