2026-03-18

Problem — 2026-03-18

Medium

Mediumprobability

You start with $\$3$ . Each round you flip a fair coin: heads wins $\$1$ , tails loses $\$1$ . The game ends when you reach $\$0$ or $\$5$ . What is the probability you finish with $\$5$ ?

Express as a fraction

Show solution

Let

p_k

be the probability of reaching

\$5

starting from

\$k

. We have

p_0 = 0

p_5 = 1

, and

p_k = \frac{1}{2}p_{k-1} + \frac{1}{2}p_{k+1}

. This recurrence has the general solution

p_k = A + Bk

. From

p_0 = 0

A = 0

. From

p_5 = 1

B = \frac{1}{5}

. So

p_3 = \frac{3}{5}