Competition problem — Combinatorics

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Combinatorics Difficulty 2.8 Junior Find the answer

Roll two fair dice once each, and let event A be {the two numbers are different}, and event B be {at least one 5 appears}. Then the probability $P(A|B)$ equals (　　)
A: $\boxed{\frac {10}{11}}$
B: $\frac {5}{11}$
C: $\frac {5}{6}$
D: $\frac {11}{36}$

Your answer

Multiple choice: answer with the letter of the option you want.

Solution

According to the definition of conditional probability, $P(A|B)$ represents the probability of event A occurring given that event B has occurred,
which means, under the condition that "at least one 5 appears", the probability that "the two numbers are different",
The number of outcomes where "at least one 5 appears" is $6\times6-5\times5=11$ ,
For "the two numbers are different" with only one 5, there are a total of $C_{2}^{1}\times5=10$ ways,
Therefore, $P(A|B)= \boxed{\frac {10}{11}}$ .
Hence, the correct answer is: A.
This question examines conditional probability. It's important to note that the calculation of this type of probability differs from others, as $P(A|B)$ represents the probability of event A occurring given that event B has occurred.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.