Calculate the conditional probability of player 2 first eating chocolate B given different scenarios of player 1's first choice | Step-by-Step Solution

Hello! This is a great probability problem that involves conditional thinking.

What We're Solving:

We need to find the probability that Player 2 will be the first person to eat chocolate B, given that there are 4 chocolates A and 1 chocolate B, and Player 1 goes first.

The Approach:

The key insight here is to think about what needs to happen for Player 2 to eat chocolate B first. This means Player 1 must NOT eat chocolate B on their turn! We'll use the concept of conditional probability and think through the different scenarios step by step.

Step-by-Step Solution:

Step 1: Understand what we have

• Total chocolates: 5 (4 type A + 1 type B)
• Players alternate turns, Player 1 goes first
• We want: Player 2 eats chocolate B before Player 1 does

Step 2: Identify the key condition For Player 2 to eat chocolate B first, Player 1 must pick a chocolate A on their first turn. If Player 1 picks chocolate B, then Player 1 ate it first, and our event is impossible!

Step 3: Calculate Player 1's probability of NOT eating chocolate B

• Player 1 has 5 chocolates to choose from
• 4 of them are chocolate A (the "good" outcomes for our goal)
• P(Player 1 picks chocolate A) = 4/5

Step 4: Given Player 1 picked chocolate A, what's left? After Player 1 eats one chocolate A:

• Remaining chocolates: 4 total (3 type A + 1 type B)
• Now it's Player 2's turn

Step 5: Calculate Player 2's probability of eating chocolate B

• Player 2 faces 4 remaining chocolates
• 1 of them is chocolate B
• P(Player 2 picks chocolate B | Player 1 picked A) = 1/4

Step 6: Combine the probabilities Since both events must happen:

• Player 1 picks A AND then Player 2 picks B
• P(Player 2 eats B first) = P(Player 1 picks A) × P(Player 2 picks B | Player 1 picked A)
• P(Player 2 eats B first) = (4/5) × (1/4) = 4/20 = 1/5

The Answer:

The probability that Player 2 will first eat chocolate B is 1/5 or 0.2 or 20%.

Memory Tip:

Think of it this way: "For Player 2 to win the chocolate B, Player 1 must 'miss' it first!" This reminds you that you need to consider what Player 1 does before you can determine Player 2's chances. In conditional probability problems like this, always ask: "What has to happen first for my desired outcome to be possible?"

Great job working through this! Conditional probability can be tricky at first, but breaking it down into "what must happen step by step" makes it much clearer. 🌟