A certain coin is tossed with probability of showing head being `'p'`. Let `'q'` denotes the probability that when the coin is tossed four times the number of heads obtained is even. Then

Options

A. (a) there is no value of `p`, if `q=(1)/(4)`
B. (b) there is exactly one value of `p`, if `q=(3)/(4)`
C. (c) there are exactly three value of `p`, if `q=(3)/(5)`
D. (d) there are exactly four value of `p`, if `q=(4)/(5)`

Explanation

To solve the problem, we need to calculate the probability \( q \) that when a coin is tossed four times, the number of heads obtained is even. The probability of getting heads in a single toss is \( p \), and the probability of getting tails is \( 1 - p \).

### Step-by-Step Solution:

1. **Identify the Possible Outcomes**:
When a coin is tossed four times, the possible outcomes for the number of heads can be 0, 1, 2, 3, or 4 heads. We are interested in the cases where the number of heads is even, which are 0, 2, and 4 heads.

2. **Calculate the Probability of Each Case**:
- **Probability of 0 Heads**:
This occurs when all tosses result in tails. The probability is:
\[ P(0 \text{ heads}) = (1 - p)^4 \]

- **Probability of 2 Heads**:
This occurs when exactly 2 out of the 4 tosses result in heads. The number of ways to choose 2 tosses from 4 is given by \( \binom{4}{2} \), and the probability is:
\[ P(2 \text{ heads}) = \binom{4}{2} p^2 (1 - p)^2 = 6 p^2 (1 - p)^2 \]

- **Probability of 4 Heads**:
This occurs when all tosses result in heads. The probability is:
\[ P(4 \text{ heads}) = p^4 \]

3. **Combine the Probabilities**:
The total probability \( q \) of getting an even number of heads is the sum of the probabilities of getting 0, 2, and 4 heads:
\[ q = P(0 \text{ heads}) + P(2 \text{ heads}) + P(4 \text{ heads}) \]
Substituting the values we calculated:
\[ q = (1 - p)^4 + 6p^2(1 - p)^2 + p^4 \]

4. **Simplify the Expression**:
We can simplify the expression for \( q \):
\[ q = (1 - p)^4 + 6p^2(1 - p)^2 + p^4 \]
This can be rewritten as:
\[ q = (1 - p)^4 + 6p^2(1 - p)^2 + p^4 \]

5. **Final Expression**:
The final expression for \( q \) can be rearranged as:
\[ q = \frac{1}{2} + \frac{1}{2}(2p - 1)^4 \]