Five coins are tossed 3200 times. The number of times getting exactly two heads is

Options

A. 600
B. 1000
C. 2000
D. 1500

Answer

To solve the problem of finding the number of times exactly two heads appear when five coins are tossed 3200 times, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Parameters**:
- Number of coins tossed (n) = 5
- Total number of tosses = 3200
- We want to find the probability of getting exactly 2 heads (r = 2).

2. **Determine the Probability of Heads and Tails**:
- The probability of getting heads (P) = 1/2
- The probability of getting tails (Q) = 1 - P = 1/2

3. **Use the Binomial Probability Formula**:
The probability of getting exactly r heads in n tosses is given by the formula:
\[ P(X = r) = \binom{n}{r} P^r Q^{n-r} \]
Substituting the values:
\[ P(X = 2) = \binom{5}{2} \left(\frac{1}{2}\right)^2 \left(\frac{1}{2}\right)^{5-2} \]

4. **Calculate the Binomial Coefficient**:
- Calculate \(\binom{5}{2}\):
\[ \binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10 \]

5. **Calculate the Probability**:
- Now substitute back into the probability formula:
\[ P(X = 2) = 10 \left(\frac{1}{2}\right)^2 \left(\frac{1}{2}\right)^3 = 10 \left(\frac{1}{2}\right)^5 = 10 \times \frac{1}{32} = \frac{10}{32} = \frac{5}{16} \]

6. **Calculate the Expected Number of Times**:
- To find the expected number of times exactly 2 heads occur in 3200 tosses:
\[ \text{Expected number} = \text{Total tosses} \times P(X = 2) = 3200 \times \frac{5}{16} \]

7. **Perform the Multiplication**:
\[ 3200 \times \frac{5}{16} = 3200 \div 16 \times 5 = 200 \times 5 = 1000 \]

8. **Final Answer**:
The number of times getting exactly two heads when five coins are tossed 3200 times is **1000**.