A player tosses a coin and score one point for every head and two points for every tail that turns up. He plays on until his score reaches or passes n. `P_(n)` denotes the probability of getting a score of exactly n.
Which of the following is not true ?

Options

A. `P_(100)gt2//3`
B. `P_(100)lt2//3`
C. `P_(100),P_(101)gt2//3`
D. None of these

Explanation

To solve the problem regarding the probability \( P(n) \) of a player scoring exactly \( n \) points by tossing a coin, we need to analyze the scoring system and derive the probability.

### Step-by-Step Solution:

1. **Understanding the Scoring System**:
- The player scores 1 point for every head (H) and 2 points for every tail (T).
- If the player tosses the coin \( k \) times, let \( h \) be the number of heads and \( t \) be the number of tails.
- The total score can be expressed as:
\[ \text{Score} = h + 2t \]
- The total number of tosses is:
\[ k = h + t \]

2. **Setting Up the Equation**:
- We want the score to equal \( n \):
\[ h + 2t = n \]
- From the equation \( k = h + t \), we can express \( h \) in terms of \( t \):
\[ h = n - 2t \]
- Substituting \( h \) into the equation for \( k \):
\[ k = (n - 2t) + t = n - t \]
- This gives us:
\[ t = n - k \]

3. **Finding Valid Combinations**:
- The values of \( t \) must be non-negative, which means:
\[ n - k \geq 0 \implies k \leq n \]
- Also, since \( h \) must be non-negative:
\[ n - 2t \geq 0 \implies t \leq \frac{n}{2} \]

4. **Counting the Outcomes**:
- The number of ways to achieve \( h \) heads and \( t \) tails in \( k \) tosses is given by the binomial coefficient:
\[ \binom{k}{h} = \binom{k}{n - 2t} \]
- The total number of outcomes for \( k \) tosses is \( 2^k \).

5. **Calculating the Probability**:
- The probability \( P(n) \) of scoring exactly \( n \) points can be expressed as:
\[ P(n) = \sum_{k=0}^{n} \binom{k}{n - 2t} \cdot \left(\frac{1}{2}\right)^k \]
- Here, \( t \) varies from \( 0 \) to \( \frac{n}{2} \) and \( k \) varies accordingly.

6. **Conclusion**:
- The final expression for \( P(n) \) gives us the probability of achieving exactly \( n \) points based on the combinations of heads and tails.