Data Scientist Interview Question #1

Probability Question #1

Problem

Two players, A and B, take turns flipping a coin with probability p of landing heads. Player A flips first, and the first player to get heads wins. What is the probability that Player A wins the game?

This question was asked in a Goldman Sachs Data Scientist interview.

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Solution Steps

Step 1: Define Probabilities
Let:

• P_A: Probability that Player A wins.

• P_B: Probability that Player B wins.

Since one of the two players must eventually win:

P_B = 1 - P_A

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Step 2: Player A's Winning Scenarios
Player A wins in two ways:

1. Immediate Win: Player A flips heads on the first turn.

   • Probability = p.

2. Game Continues: Player A flips tails (with probability 1 - p), and then Player B gets a turn.

   • If the game resets after Player B flips tails, Player A essentially starts again.

   • From this point, the probability that Player A eventually wins is P_B = 1 - P_A.

Combining these cases, the total probability for P_A is:

P_A = p + (1 - p)(1 - P_A)

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Step 3: Solve for P_A
Start with the equation:

P_A = p + (1 - p)(1 - P_A)

Simplify the terms:

P_A = p + (1 - p) - (1 - p)P_A

Combine like terms:

P_A + (1 - p)P_A = p + (1 - p)

Factor out P_A:

P_A * (1 - p) = 1

Solve for P_A:

P_A = p / (1 - p)

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Final Answer

The probability that Player A wins is:

P_A = p / (1 - p)

Player A has an advantage because they flip first. The recursive nature of the game (flipping tails resets the process) creates a geometric series, leading to the solution.