medanova wrote:

Two cards are drawn at random from a pack of 52 cards. What is the probability that either both are black or both are queens?

Is there a way to solve it without using the combinations formula?

Yes, there is.

OR probability: If Events A and B are independent, the probability that either Event A OR Event B occurs is: \(P(A \ or \ B) = P(A) + P(B) - P(A \ and \ B)\).

This is basically the same as 2 overlapping sets formula:

{total # of items in groups A or B} = {# of items in group A} + {# of items in group B} - {# of items in A and B}.

Note that if event are mutually exclusive then \(P(A \ and \ B)=0\) and the formula simplifies to: \(P(A \ or \ B) = P(A) + P(B)\).

Also note that when we say "A or B occurs" we include three possibilities:

A occurs and B does not occur;

B occurs and A does not occur;

Both A and B occur.

AND probability:When two events are independent, the probability of both occurring is the product of the probabilities of the individual events: \(P(A \ and \ B) = P(A)*P(B)\).

This is basically the same as

Principle of Multiplication: if one event can occur in \(m\) ways and a second can occur independently of the first in \(n\) ways, then the two events can occur in \(mn\) ways.

Back to the original question:Two cards are drawn at random from a pack of 52 cards. What is the probability that either both are black or both are queens?

\(P(BB \ or QQ)=P(BB)+P(QQ)-P(BQ \ and \ BQ)=\frac{26}{52}*\frac{25}{51}+\frac{4}{52}*\frac{3}{51}-\frac{2}{52}*\frac{1}{51}\) (note \(P(BQ \ and \ BQ)\) means the probability that first card is black queen and the second card is also black queen).

Hope its clear.

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