Two coins are tossed five times. The probability that an odd number of heads are obtained, is

Options

A. `((1)/(2))^5`
B. `(3)/(5)`
C. `(2)/(5)`
D. none of these

Answer

To solve the problem of finding the probability of obtaining an odd number of heads when two coins are tossed five times, we can follow these steps:

### Step 1: Understand the Experiment
When two coins are tossed, the possible outcomes for each toss are:
- HH (2 heads)
- HT (1 head)
- TH (1 head)
- TT (0 heads)

Thus, there are 4 possible outcomes for each toss.

### Step 2: Determine the Probability of Heads
The probability of getting heads (H) in a single toss of two coins can be calculated as follows:
- Probability of getting 0 heads (TT) = 1/4
- Probability of getting 1 head (HT or TH) = 2/4 = 1/2
- Probability of getting 2 heads (HH) = 1/4

Thus, the probability of getting at least one head in a single toss is:
- P(H) = Probability of getting 1 head + Probability of getting 2 heads = 1/2 + 1/4 = 3/4
- Probability of getting tails (T) = 1 - P(H) = 1/4

### Step 3: Define the Random Variable
Let X be the random variable representing the number of heads obtained in 5 tosses of two coins. We want to find the probability of X being odd (1, 3, or 5 heads).

### Step 4: Use the Binomial Probability Formula
The number of heads follows a binomial distribution, where:
- n = number of trials = 5 (since we toss the coins 5 times)
- p = probability of getting heads in one toss = 3/4
- q = probability of getting tails in one toss = 1/4

The probability of getting exactly r heads in n tosses is given by:
\[ P(X = r) = \binom{n}{r} p^r q^{n-r} \]

### Step 5: Calculate the Required Probabilities
We need to calculate:
- \( P(X = 1) \)
- \( P(X = 3) \)
- \( P(X = 5) \)

1. **Calculate \( P(X = 1) \)**:
\[ P(X = 1) = \binom{5}{1} \left(\frac{3}{4}\right)^1 \left(\frac{1}{4}\right)^{4} = 5 \cdot \frac{3}{4} \cdot \frac{1}{256} = \frac{15}{1024} \]

2. **Calculate \( P(X = 3) \)**:
\[ P(X = 3) = \binom{5}{3} \left(\frac{3}{4}\right)^3 \left(\frac{1}{4}\right)^{2} = 10 \cdot \left(\frac{27}{64}\right) \cdot \left(\frac{1}{16}\right) = \frac{270}{1024} \]

3. **Calculate \( P(X = 5) \)**:
\[ P(X = 5) = \binom{5}{5} \left(\frac{3}{4}\right)^5 \left(\frac{1}{4}\right)^{0} = 1 \cdot \frac{243}{1024} = \frac{243}{1024} \]

### Step 6: Sum the Probabilities
Now, we sum the probabilities of getting an odd number of heads:
\[ P(X \text{ is odd}) = P(X = 1) + P(X = 3) + P(X = 5) = \frac{15}{1024} + \frac{270}{1024} + \frac{243}{1024} = \frac{528}{1024} \]

### Step 7: Simplify the Result
\[ P(X \text{ is odd}) = \frac{528}{1024} = \frac{33}{64} \]

### Conclusion
The probability that an odd number of heads are obtained when two coins are tossed five times is \( \frac{33}{64} \).

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