Exercise 2.5

Question

Three cards are dealt from a standard, well-shuffled deck. The first two cards are flipped
over, revealing the Ace of Spades as the first card and the 8 of Clubs as the second card.
Given this information, find the probability that the third card is an ace in two ways: using the definition of conditional probability, and by symmetry.

fifthist x · Accepted Answer

By symmetry, all 50 of the remaining cards are equally likely. Thus, the probability that the third card is an ace is $\frac{3}{50}$.

We can reach the same answer using the definition of conditional probability. 
Let $A$ be the event that the first card is the Ace of Spades, $B$ be the event 
that the second card is the $8$ of Clubs and $C$ be the event that the third 
card is an ace. Then, 
$$P(C|A,B) = \dfrac{P(C,A,B)}{P(A,B)} = \dfrac{\ \dfrac{3*49!}{52!}\ }{\dfrac{50!}{52!}} 
= \frac{3}{50}$$

Exercise 2.5

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Exercise 2.5

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