A and B play a game where each is asked to select a number from 1 to 25. If the two numbers match, both of them win a prize. The probability that they will not win a prize in a single trial is

Options

A. `1/25`
B. `24/25`
C. `2/25`
D. None of these

Answer

To solve the problem, we need to calculate the probability that A and B will not win a prize in a single trial when each selects a number from 1 to 25.

### Step-by-Step Solution:

1. **Determine the Total Number of Outcomes:**
- Each player (A and B) can choose any number from 1 to 25.
- Therefore, the total number of outcomes when both players select a number is:
\[ \text{Total Outcomes} = 25 \times 25 = 625 \]

**Hint:** Think of it as a grid where A's choices are on one axis and B's choices on the other.

2. **Determine the Favorable Outcomes for Winning:**
- A and B win a prize if they select the same number. There are 25 numbers they can both choose from (1 to 25).
- Thus, the number of favorable outcomes for winning (where both numbers match) is:
\[ \text{Favorable Outcomes for Winning} = 25 \]

**Hint:** Each number from 1 to 25 represents a winning combination.

3. **Calculate the Probability of Winning:**
- The probability that A and B will win a prize is given by the ratio of favorable outcomes to total outcomes:
\[ P(\text{Winning}) = \frac{\text{Favorable Outcomes for Winning}}{\text{Total Outcomes}} = \frac{25}{625} = \frac{1}{25} \]

**Hint:** Remember that probability is always calculated as favorable outcomes divided by total outcomes.

4. **Calculate the Probability of Not Winning:**
- The probability that A and B will not win a prize is the complement of the probability of winning:
\[ P(\text{Not Winning}) = 1 - P(\text{Winning}) = 1 - \frac{1}{25} = \frac{24}{25} \]

**Hint:** The complement rule states that the probability of an event not occurring is 1 minus the probability of the event occurring.

### Final Answer:
The probability that A and B will not win a prize in a single trial is:
\[ \frac{24}{25} \]