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Difficulty: HardProbability of Independent and Dependent Events

A game features two boxes of tokens. Box X contains 22 red tokens and 33 blue tokens. Box Y contains 44 red tokens and 11 blue token. A fair coin is flipped to determine which box to draw from: if the coin lands on heads, Box X is chosen; if it lands on tails, Box Y is chosen. Two tokens are then drawn sequentially without replacement from the chosen box. What is the probability that both drawn tokens are red?

Answer: 0.35

Answer

The probability that both drawn tokens are red is 0.35.
The total probability combines the independent choice of box with dependent draws without replacement. Box X yields two red tokens with probability 0.100.10, and Box Y yields two red tokens with probability 0.600.60. Weighting each by the 0.50.5 probability of choosing that box gives (0.5×0.10)+(0.5×0.60)=0.35(0.5 \times 0.10) + (0.5 \times 0.60) = 0.35.

Step-by-Step Solution

1
Determine the conditional probability of drawing two red tokens from Box X without replacement.
The probability is 25×14=0.10\frac{2}{5} \times \frac{1}{4} = 0.10.
Because draws are dependent (without replacement), the number of remaining red tokens decreases to 1 and total tokens to 4 after the first red draw.
2
Determine the conditional probability of drawing two red tokens from Box Y without replacement.
The probability is 45×34=0.60\frac{4}{5} \times \frac{3}{4} = 0.60.
Box Y initially contains 4 red out of 5 total tokens; drawing one red leaves 3 red out of 4 total tokens.
3
Combine the independent box selection probabilities with the dependent drawing probabilities.
Overall probability is (0.5×0.10)+(0.5×0.60)=0.05+0.30=0.35(0.5 \times 0.10) + (0.5 \times 0.60) = 0.05 + 0.30 = 0.35.
The initial coin flip selects Box X or Box Y with equal, independent probability of 0.5.

Key Concept

Combining independent events (environment selection) with dependent events (sampling without replacement) using the Law of Total Probability.
Estimated Time:2m 0s
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Topics

Probability of Independent and Dependent Events

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