A and B toss 3 coins . The probability that they both obtain the same number of heads is

Options

A. `1/9`
B. `3/16`
C. `5/16`
D. `3/8`

Answer

To solve the problem of finding the probability that A and B both obtain the same number of heads when tossing 3 coins, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
- A and B each toss 3 coins. We need to find the probability that they both get the same number of heads.

2. **Possible Outcomes**:
- When tossing 3 coins, the possible numbers of heads (0, 1, 2, or 3) can be represented as:
- 0 heads
- 1 head
- 2 heads
- 3 heads

3. **Using Binomial Distribution**:
- The number of heads obtained when tossing 3 coins can be modeled using the binomial distribution. The probability of getting exactly \( k \) heads in \( n \) tosses is given by:
\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]
- Here, \( n = 3 \) (number of tosses), \( p = \frac{1}{2} \) (probability of getting heads), and \( 1-p = \frac{1}{2} \) (probability of getting tails).

4. **Calculating Probabilities for Each Case**:
- We will calculate the probability for both A and B getting the same number of heads (0, 1, 2, or 3 heads).

- **Case 1**: Both get 0 heads:
\[ P(X = 0) = \binom{3}{0} \left(\frac{1}{2}\right)^0 \left(\frac{1}{2}\right)^3 = 1 \cdot 1 \cdot \frac{1}{8} = \frac{1}{8} \]

- **Case 2**: Both get 1 head:
\[ P(X = 1) = \binom{3}{1} \left(\frac{1}{2}\right)^1 \left(\frac{1}{2}\right)^2 = 3 \cdot \frac{1}{2} \cdot \frac{1}{4} = \frac{3}{8} \]

- **Case 3**: Both get 2 heads:
\[ P(X = 2) = \binom{3}{2} \left(\frac{1}{2}\right)^2 \left(\frac{1}{2}\right)^1 = 3 \cdot \frac{1}{4} \cdot \frac{1}{2} = \frac{3}{8} \]

- **Case 4**: Both get 3 heads:
\[ P(X = 3) = \binom{3}{3} \left(\frac{1}{2}\right)^3 \left(\frac{1}{2}\right)^0 = 1 \cdot \frac{1}{8} \cdot 1 = \frac{1}{8} \]

5. **Total Probability of Same Heads**:
- Now, we sum the probabilities of all cases where A and B get the same number of heads:
\[ P(\text{same heads}) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) \]
\[ P(\text{same heads}) = \frac{1}{8} + \frac{3}{8} + \frac{3}{8} + \frac{1}{8} = \frac{8}{8} = 1 \]

6. **Final Calculation**:
- Since A and B are independent, the combined probability that they both get the same number of heads is:
\[ P(\text{same heads}) = \left(\frac{1}{8}\right)^2 + \left(\frac{3}{8}\right)^2 + \left(\frac{3}{8}\right)^2 + \left(\frac{1}{8}\right)^2 \]
\[ = \frac{1}{64} + \frac{9}{64} + \frac{9}{64} + \frac{1}{64} = \frac{20}{64} = \frac{5}{16} \]

### Conclusion:
The probability that A and B both obtain the same number of heads when tossing 3 coins is \( \frac{5}{16} \).