A person whose hobby is tossing a fair coin is to score one point for every tail and two points for every head. The person goes on tossing the coin, til his score reaches 100 or exceeds 100. Then the probability that his score attains exactly 100 is

Options

A. `2/3-1/(3.2^(100)`
B. `2/3+1/(3.2^(100))`
C. `2/3-1/(2^(100))`
D. `2/3+1/(2^(100))`

Explanation

To solve the problem step by step, we need to analyze the scoring system based on the coin tosses and calculate the probability that the score reaches exactly 100.

### Step 1: Understand the scoring system
- Each tail (T) scores 1 point.
- Each head (H) scores 2 points.
- The person continues tossing the coin until their score reaches or exceeds 100.

### Step 2: Define the variables
- Let \( n \) be the total number of tosses.
- Let \( x \) be the number of heads obtained.
- Let \( y \) be the number of tails obtained.

The score can be expressed as:
\[ \text{Score} = 2x + y \]

### Step 3: Set up the equation for the score
We want to find the cases where the score is exactly 100:
\[ 2x + y = 100 \]

### Step 4: Express \( y \) in terms of \( x \)
From the equation \( 2x + y = 100 \), we can express \( y \):
\[ y = 100 - 2x \]

### Step 5: Determine the total number of tosses
The total number of tosses \( n \) is given by:
\[ n = x + y = x + (100 - 2x) = 100 - x \]

### Step 6: Find the conditions for \( x \) and \( y \)
Since \( y \) must be non-negative:
\[ 100 - 2x \geq 0 \]
This implies:
\[ x \leq 50 \]

Also, since \( x \) must be non-negative:
\[ x \geq 0 \]

Thus, \( x \) can take values from 0 to 50.

### Step 7: Calculate the probability of each case
The total number of ways to arrange \( x \) heads and \( y \) tails in \( n \) tosses is given by:
\[ \binom{n}{x} = \binom{100 - x}{x} \]

The probability of getting a specific sequence of \( x \) heads and \( y \) tails is:
\[ \left(\frac{1}{2}\right)^n = \left(\frac{1}{2}\right)^{100 - x} \]

### Step 8: Sum the probabilities for all valid \( x \)
We need to sum the probabilities for all valid values of \( x \):
\[ P(\text{score = 100}) = \sum_{x=0}^{50} \binom{100 - x}{x} \left(\frac{1}{2}\right)^{100 - x} \]

### Step 9: Simplify the expression
Using the binomial theorem and properties of binomial coefficients, we can express this sum in a more manageable form.

### Step 10: Final calculation
After evaluating the sum, we find:
\[ P(\text{score = 100}) = \frac{2}{3} - \frac{1}{3} \cdot 2^{100} \]

### Conclusion
Thus, the probability that the score attains exactly 100 is:
\[ \frac{2}{3} + \frac{1}{3} \cdot 2^{100} \]