Exercise 12 Page 929

We want to determine whether the events are mutually exclusive and find the probability of their union. We will do these things one at a time.

Mutually Exclusive

Events that cannot occur at the same time are mutually exclusive. Mutually exclusive events have no outcomes in common. For example, because it is not possible to toss a coin and obtain heads and tails at the same time, these two events are mutually exclusive.

One out of the four aces is a heart, so it is possible to be both an ace and a heart in the given deck. Therefore, A and B are not mutually exclusive events.

Probability

Now, we can find the probability that the card drawn is either an ace or a heart.

Addition Rules for Probability
If A and B are mutually exclusive events, the probability that A or B will occur is P(AorB)=P(A)+P(B).	If A and B are not mutually exclusive events, the probability that A or B will occur is P(AorB)=P(A)+P(B)-P(AandB).

We know that there are 52 cards in the deck. Out of these, 4 cards are aces, 13 cards have a heart suit, and only 1 card is the ace of hearts. Next, we can calculate P(A), P(B), and P(AandB). P(A)&=4/52 l← ← lAces Total cards [0.5em] P(B)&=13/52 l← ← lHearts Total cards [0.5em] P(AandB)&=1/52 l← ← lAce of hearts Total cards Finally, considering that A and B are not mutually exclusive events, let's calculate P(AorB).

P(AorB)=P(A)+P(B)-P(AandB)

SubstituteValues

Substitute values

P(AorB)=4/52+ 13/52- 1/52

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Simplify right-hand side

AddSubFrac

Add and subtract fractions

P(AorB)=16/52

ReduceFrac

a/b=.a /4./.b /4.

P(AorB)=4/13

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Convert to percent

CalcQuot

Calculate quotient

P(AorB)=0.307692...

RoundDec

Round to 3 decimal place(s)

P(AorB)≈0.308

WritePercent

Convert to percent

P(AorB)≈30.8 %

We found that P(ace or heart) is equal to 413 or about 30.8 %.