Each of A and B tosses two coins. What is the probability that they get equal number of heads ?

Options

A. `3/16`
B. `5/16`
C. `4/16`
D. `6/16`

Explanation

To solve the problem of finding the probability that A and B get an equal number of heads when each tosses two coins, we can follow these steps:

### Step 1: Determine the Sample Space
When tossing two coins, the possible outcomes for each toss can be represented as:
- HH (2 heads)
- HT (1 head, 1 tail)
- TH (1 head, 1 tail)
- TT (0 heads)

Thus, the sample space for tossing two coins is:
\[ S = \{ HH, HT, TH, TT \} \]
This gives us a total of 4 outcomes.

### Step 2: Calculate the Probability of Each Outcome
The probabilities of getting 0, 1, or 2 heads when tossing two coins can be calculated as follows:
- Probability of getting 0 heads (TT): \( P(0) = \frac{1}{4} \)
- Probability of getting 1 head (HT or TH): \( P(1) = \frac{2}{4} = \frac{1}{2} \)
- Probability of getting 2 heads (HH): \( P(2) = \frac{1}{4} \)

### Step 3: Identify Equal Outcomes for A and B
Next, we need to find the scenarios where A and B have an equal number of heads. The possible cases are:
1. Both get 0 heads.
2. Both get 1 head.
3. Both get 2 heads.

### Step 4: Calculate the Probabilities for Each Case
1. **Both get 0 heads:**
\[ P(A = 0 \text{ and } B = 0) = P(0) \times P(0) = \frac{1}{4} \times \frac{1}{4} = \frac{1}{16} \]

2. **Both get 1 head:**
\[ P(A = 1 \text{ and } B = 1) = P(1) \times P(1) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} = \frac{4}{16} \]

3. **Both get 2 heads:**
\[ P(A = 2 \text{ and } B = 2) = P(2) \times P(2) = \frac{1}{4} \times \frac{1}{4} = \frac{1}{16} \]

### Step 5: Sum the Probabilities
Now, we sum the probabilities of the cases where A and B have equal heads:
\[ P(\text{equal heads}) = P(A = 0 \text{ and } B = 0) + P(A = 1 \text{ and } B = 1) + P(A = 2 \text{ and } B = 2) \]
\[ P(\text{equal heads}) = \frac{1}{16} + \frac{4}{16} + \frac{1}{16} = \frac{6}{16} = \frac{3}{8} \]

### Step 6: Final Probability
Thus, the probability that A and B get an equal number of heads is:
\[ \frac{3}{8} \]