Two persons each makes a single throw with a die. The probability they get equal value is `P_(1)`. Four persons each makes a single throw and probability of three being equal is `P_(1)`. 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 \( P_2 \) and then compare them.

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

**Explanation:**
When two persons each throw a die, we want to find the probability that they get the same value.

1. The total number of outcomes when two dice are thrown is \( 6 \times 6 = 36 \).
2. The favorable outcomes for them to get the same value are: (1,1), (2,2), (3,3), (4,4), (5,5), (6,6). There are 6 favorable outcomes.

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

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

**Explanation:**
Now, we have four persons each throwing a die, and we want to find the probability that exactly three of them get the same value.

1. To find this probability, we first choose which three persons will have the same value. This can be done in \( \binom{4}{3} = 4 \) ways.
2. The value that these three persons will show can be any one of the 6 values (1 through 6).
3. The fourth person must show a different value, which can be any of the remaining 5 values.

**Calculation:**
- The number of favorable outcomes:
\[ \text{Favorable outcomes} = \binom{4}{3} \times 6 \times 5 = 4 \times 6 \times 5 = 120 \]
- The total number of outcomes when four dice are thrown is \( 6^4 = 1296 \).

Thus,
\[ P_2 = \frac{120}{1296} = \frac{5}{54} \]

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

**Calculation:**
Now we have:
- \( P_1 = \frac{1}{6} \)
- \( P_2 = \frac{5}{54} \)

To compare \( P_1 \) and \( P_2 \), we can convert \( P_1 \) to a fraction with a denominator of 54:
\[ P_1 = \frac{1}{6} = \frac{9}{54} \]

Now we compare:
\[ P_1 = \frac{9}{54} \quad \text{and} \quad P_2 = \frac{5}{54} \]

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

### Final Answer:
Thus, the relation is \( P_1 > P_2 \).

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