Question

given a standard deck of cards, what is the probability of randomly choosing an eight given that it is a black card? simplified

Explanation:

Step1: Determine the number of black cards and eight - of - black cards

In a standard deck of 52 cards, there are 26 black cards. Among the black cards, there are 2 eight - of - black cards (eight of spades and eight of clubs).

Step2: Use the formula for conditional probability

The formula for conditional probability \(P(A|B)=\frac{P(A\cap B)}{P(B)}\). In the case of conditional probability for equally - likely outcomes, if \(A\) is the event of choosing an eight and \(B\) is the event of choosing a black card, then \(P(A|B)=\frac{n(A\cap B)}{n(B)}\), where \(n(A\cap B)\) is the number of elements in the intersection of \(A\) and \(B\), and \(n(B)\) is the number of elements in \(B\). Here, \(n(A\cap B) = 2\) (the two black eights) and \(n(B)=26\).

Step3: Simplify the fraction

\(P=\frac{2}{26}=\frac{1}{13}\)

Answer:

\(\frac{1}{13}\)