A coin is tossed 3 times by 2 persons. The prbability that both get equal number of heads, is

Options

A. `3//8`
B. `1//9`
C. `5//16`
D. none of these

Explanation

To solve the problem of finding the probability that both persons get an equal number of heads when a coin is tossed three times by each, we can follow these steps:

### Step 1: Understand the Total Outcomes
When a coin is tossed three times, the possible outcomes for the number of heads (H) can be 0, 1, 2, or 3. The total number of outcomes when tossing a coin three times is \(2^3 = 8\).

### Step 2: Calculate the Probability of Each Outcome
We can calculate the probability of getting 0, 1, 2, or 3 heads using the binomial probability formula:
\[ P(H = k) = \binom{n}{k} \left(\frac{1}{2}\right)^n \]
where \(n\) is the number of tosses (3 in this case), and \(k\) is the number of heads.

- **Probability of getting 0 heads:**
\[ P(H = 0) = \binom{3}{0} \left(\frac{1}{2}\right)^3 = 1 \cdot \frac{1}{8} = \frac{1}{8} \]

- **Probability of getting 1 head:**
\[ P(H = 1) = \binom{3}{1} \left(\frac{1}{2}\right)^3 = 3 \cdot \frac{1}{8} = \frac{3}{8} \]

- **Probability of getting 2 heads:**
\[ P(H = 2) = \binom{3}{2} \left(\frac{1}{2}\right)^3 = 3 \cdot \frac{1}{8} = \frac{3}{8} \]

- **Probability of getting 3 heads:**
\[ P(H = 3) = \binom{3}{3} \left(\frac{1}{2}\right)^3 = 1 \cdot \frac{1}{8} = \frac{1}{8} \]

### Step 3: Calculate the Probability of Both Getting the Same Number of Heads
Now we need to find the probability that both persons get the same number of heads. This can happen in the following scenarios:

1. Both get 0 heads.
2. Both get 1 head.
3. Both get 2 heads.
4. Both get 3 heads.

The probability for each scenario is calculated as follows:

- **Both get 0 heads:**
\[ P(0, 0) = P(H = 0) \times P(H = 0) = \left(\frac{1}{8}\right) \times \left(\frac{1}{8}\right) = \frac{1}{64} \]

- **Both get 1 head:**
\[ P(1, 1) = P(H = 1) \times P(H = 1) = \left(\frac{3}{8}\right) \times \left(\frac{3}{8}\right) = \frac{9}{64} \]

- **Both get 2 heads:**
\[ P(2, 2) = P(H = 2) \times P(H = 2) = \left(\frac{3}{8}\right) \times \left(\frac{3}{8}\right) = \frac{9}{64} \]

- **Both get 3 heads:**
\[ P(3, 3) = P(H = 3) \times P(H = 3) = \left(\frac{1}{8}\right) \times \left(\frac{1}{8}\right) = \frac{1}{64} \]

### Step 4: Sum the Probabilities
Now, we sum all the probabilities of both persons getting the same number of heads:
\[ P(\text{same heads}) = P(0, 0) + P(1, 1) + P(2, 2) + P(3, 3) \]
\[ P(\text{same heads}) = \frac{1}{64} + \frac{9}{64} + \frac{9}{64} + \frac{1}{64} = \frac{20}{64} = \frac{5}{16} \]

### Conclusion
Thus, the probability that both persons get an equal number of heads is \(\frac{5}{16}\).

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