First slide

Introduction to probability

Question

A draws a card from a pack of n cards marked 1,2, ..., n. The card is replaced in the pack and B draws a card. Then the probability that A draws a higher card than B is

Moderate

A

$(n + 1) / 2 n$
B

1/2
C

$(n - 1) / 2 n$
D

none of these

Solution

If A draws card higher than B, then the number of favorable cases is (n - 1) + (n - 2) +... + 3 + 2 + 1 (as when B draws card number 1, then A can draw any card from 2 to n and so on). Therefore, the required probability is
$\frac{\frac{n (n - 1)}{2}}{n^{2}} = \frac{n - 1}{2 n}$

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