Probability Explanation And Exercises Worksheet Page 8
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37Probability of A and B
When two events are independent, the probability of both occurring is the product
of the probabilities of the individual events. More formally, if events A and B are
independent, then the probability of both A and B occurring is:
P(A and B) = P(A) x P(B)
where P(A and B) is the probability of events A and B both occurring, P(A) is the
probability of event A occurring, and P(B) is the probability of event B occurring.
If you flip a coin twice, what is the probability that it will come up heads
both times? Event A is that the coin comes up heads on the first flip and Event B is
that the coin comes up heads on the second flip. Since both P(A) and P(B) equal
1/2, the probability that both events occur is
1/2 x 1/2 = 1/4
Let’s take another example. If you flip a coin and roll a six-sided die, what is
the probability that the coin comes up heads and the die comes up 1? Since the two
events are independent, the probability is simply the probability of a head (which is
1/2) times the probability of the die coming up 1 (which is 1/6). Therefore, the
probability of both events occurring is 1/2 x 1/6 = 1/12.
One final example: You draw a card from a deck of cards, put it back, and
then draw another card. What is the probability that the first card is a heart and the
second card is black? Since there are 52 cards in a deck and 13 of them are hearts,
the probability that the first card is a heart is 13/52 = 1/4. Since there are 26 black
cards in the deck, the probability that the second card is black is 26/52 = 1/2. The
probability of both events occurring is therefore 1/4 x 1/2 = 1/8.
See the discussion on conditional probabilities on this page to see how to
compute P(A and B) when A and B are not independent.
Probability of A or B
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)
In this discussion, when we say “A or B occurs” we include three possibilities:
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