Probability Question - None

Question

For Problems 19–21: A player tosses a coin and scores 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 $n$.

The value of $P_n$ is equal to

Accepted Answer

Step 1: Define base cases. The player starts at score 0, so P_0 = 1. With one toss: getting exactly 1 point requires a head (probability 1/2), so P_1 = 1/2. Getting exactly 2 points requires a tail (probability 1/2), so P_2 = 1/2.

Step 2: For n ≥ 3, establish recurrence relation. To reach exactly score n, the player must either:

Be at score (n-1) and toss a head (1 point): contributes (1/2)P_{n-1}
Be at score (n-2) and toss a tail (2 points): contributes (1/2)P_{n-2}

Therefore: P_n = (1/2)P_{n-1} + (1/2)P_{n-2}

Step 3: Solve the recurrence. The characteristic equation is r² = (1/2)r + (1/2), giving 2r² - r - 1 = 0, so (2r + 1)(r - 1) = 0. Roots are r = 1 and r = -1/2.

General solution: P_n = A(1)ⁿ + B(-1/2)ⁿ = A + B(-1/2)ⁿ

Step 4: Apply initial conditions. From P_0 = 1: A + B = 1. From P_1 = 1/2: A - B/2 = 1/2. Solving: A = 2/3, B = 1/3.

∴ P_n = 2/3 + (1/3)(-1/2)ⁿ or equivalently P_n = (2^{n+1} + (-1)^n)/(3·2ⁿ)

Question:

Step-by-Step Solution

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