Two persons each make a single throw with a die. The probability they get equal value is `P_(1)`. Four persons each make a single throw and probability of three being equal is `P_(2)`. Then

Options

A. `P_(1)=P_(2)`
B. `P_(1) lt P_(2)`
C. `P_(1) gt P_(2)`
D. none

Answer

To solve the problem, we need to find the probabilities \( P_1 \) and \( P_2 \) as described in the question.

### Step 1: Calculate \( P_1 \)

**Situation**: Two persons each throw a die. We need to find the probability that they get the same value.

**Total outcomes when throwing two dice**:
- Each die has 6 faces, so the total outcomes when throwing two dice is:
\[ 6 \times 6 = 36 \]

**Favorable outcomes for equal values**:
- The pairs that yield equal values are (1,1), (2,2), (3,3), (4,4), (5,5), and (6,6). Thus, there are 6 favorable outcomes.

**Calculating \( P_1 \)**:
\[ P_1 = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{6}{36} = \frac{1}{6} \]

### Step 2: Calculate \( P_2 \)

**Situation**: Four persons each throw a die. We need to find the probability that exactly three of them get the same value.

**Total outcomes when throwing four dice**:
- The total outcomes when throwing four dice is:
\[ 6^4 = 1296 \]

**Favorable outcomes for three being equal**:
1. **Choose the number that appears three times**: There are 6 choices (1 through 6).
2. **Choose the number that appears once**: There are 5 remaining choices (since it must be different from the first).
3. **Choose the position of the different number**: We can choose 1 out of the 4 positions for the different number, which can be done in \( \binom{4}{1} = 4 \) ways.

**Calculating the total favorable outcomes**:
\[ \text{Total favorable outcomes} = 6 \times 5 \times 4 = 120 \]

**Calculating \( P_2 \)**:
\[ P_2 = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{120}{1296} = \frac{5}{54} \]

### Step 3: Compare \( P_1 \) and \( P_2 \)

Now we compare \( P_1 \) and \( P_2 \):
- \( P_1 = \frac{1}{6} \)
- \( P_2 = \frac{5}{54} \)

To compare these fractions, we can convert them to a common denominator or compare their decimal values:
- \( \frac{1}{6} = \frac{9}{54} \)

Since \( \frac{9}{54} > \frac{5}{54} \), we conclude that:
\[ P_1 > P_2 \]

### Final Answer
Thus, \( P_1 \) is greater than \( P_2 \).

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