Two cards are drawn at random and without replacement from a pack of 52 playing cards. Find the probability that both the cards are black.
Solution and Explanation
S = 52 cards
⇒ 52 Two cards are drawn without replacement.
A = {26 black cards}
⇒ n(A) = 26
P(A) = \(\frac {26}{52}\)
And P(B) i.e., the probability that the second card is black known that the first card is black = \(\frac {25}{51}\)
P(A and B) = P(A).P(B)
= \(\frac {26}{52} ×\frac {25}{51}\)
= \(\frac {1}{2} ×\frac {25}{51}\)
= \(\frac {25}{102}\)
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Concepts Used:
Multiplication Theorem on Probability
In accordance with the multiplication rule of probability, the probability of happening of both the events A and B is equal to the product of the probability of B occurring and the conditional probability that event A happens given that event B occurs.
Let's assume, If A and B are dependent events, then the probability of both events occurring at the same time is given by:
\(P(A\cap B) = P(B).P(A|B)\)
Let's assume, If A and B are two independent events in an experiment, then the probability of both events occurring at the same time is given by:
\(P(A \cap B) = P(A).P(B)\)
Read More: Multiplication Theorem on Probability