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Probability of two events when a card is replaced

Replacement resets the sample space

If the first card is returned before the second draw, each draw uses the original set. The events are independent, so multiply their probabilities. Without replacement, the second denominator and possibly its numerator would change.

Problem

Cards numbered $1$ through $8$ are in a stack. Draw one at random, replace it, then draw again. What is the probability that the first card is even and the second is greater than $6$ ? Give a simplified fraction.

Answer

\frac{1}{8}

Replacement keeps all eight cards available for the second draw.

Step-by-step solution

1. First draw

Four of the eight cards are even: $2, 4, 6, 8$ . Thus $P (even first) = 4/8 = 1/2$ .

2. Second draw

After replacement, all eight cards are present again. Two, $7$ and $8$ , are greater than $6$ , so $P (> 6 second) = 2/8 = 1/4$ .

3. Multiply

The draws are independent because the card was replaced:

P (even first and > 6 second) = \frac{1}{2} \cdot \frac{1}{4} = \frac{1}{8} .

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