Basics Of Probability

Step 1: List all possible outcomes.

A standard die has six faces, numbered . Thus, the total number of outcomes is 6.

Step 2: Identify the prime numbers among the outcomes.

A prime number is a number greater than 1 that has no divisors other than 1 and itself. Among the numbers , the prime numbers are:

There are 3 prime numbers.

Step 3: Compute the probability.

The probability is the ratio of favorable outcomes (prime numbers) to total outcomes:

Simplify:

Final Answer: The probability of getting a prime number is , which corresponds to option .

Step 1: List all possible outcomes.

When two coins are tossed, the possible outcomes are:

where represents heads and represents tails. Thus, there are 4 possible outcomes.

Step 2: Identify favorable outcomes for "at least one head".

The event "at least one head" includes all outcomes where one or both coins show heads. These outcomes are:

There are 3 favorable outcomes.

Step 3: Compute the probability.

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

Final Answer: The probability of getting at least one head is , which corresponds to option .

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:

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:

Step 4: Compute the probability.

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

Simplify:

Final Answer: The probability that the two friends have the same birthday is , which corresponds to option .