Two persons A and B throw a coin alternatively till one of them gets head and wins the game. Find their respective probabilities of winning.

Options

A. `1/3 ,5/6`
B. `3/5 , 4/5`
C. `2/3 , 1/3`
D. `1/6, 5/6`

Answer

To find the respective probabilities of winning for persons A and B when they throw a coin alternatively until one of them gets a head, we can follow these steps:

### Step 1: Understand the Game
- A and B take turns throwing a coin.
- The game continues until one of them gets a head.
- The probability of getting a head (H) is \( \frac{1}{2} \) and the probability of getting a tail (T) is also \( \frac{1}{2} \).

### Step 2: Determine Winning Scenarios for A
- A can win in the following ways:
1. A gets a head on the first toss.
2. A gets a tail on the first toss, B gets a tail on the second toss, and then A gets a head on the third toss.
3. A gets a tail on the first toss, B gets a tail on the second toss, A gets a tail on the third toss, B gets a tail on the fourth toss, and then A gets a head on the fifth toss.

This pattern continues indefinitely.

### Step 3: Write the Probability of A Winning
- The probability of A winning can be expressed as:
\[ P(A) = P(H) + P(T) \cdot P(T) \cdot P(H) + P(T) \cdot P(T) \cdot P(T) \cdot P(T) \cdot P(H) + \ldots \]
- This can be simplified to:
\[ P(A) = \frac{1}{2} + \left(\frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2}\right) P(A) \]
where the term \( P(A) \) appears again because the scenario resets after both A and B get tails.

### Step 4: Set Up the Equation
- Let \( P(A) \) be the probability of A winning. We can express this as:
\[ P(A) = \frac{1}{2} + \frac{1}{4} P(A) \]

### Step 5: Solve for \( P(A) \)
- Rearranging the equation gives:
\[ P(A) - \frac{1}{4} P(A) = \frac{1}{2} \]
\[ \frac{3}{4} P(A) = \frac{1}{2} \]
- Multiplying both sides by \( \frac{4}{3} \):
\[ P(A) = \frac{1}{2} \cdot \frac{4}{3} = \frac{2}{3} \]

### Step 6: Find the Probability of B Winning
- Since there are only two players, the probability of B winning is:
\[ P(B) = 1 - P(A) = 1 - \frac{2}{3} = \frac{1}{3} \]

### Conclusion
- The probabilities of winning are:
- Probability of A winning: \( \frac{2}{3} \)
- Probability of B winning: \( \frac{1}{3} \)