Lauretta J. Fox
Two or more events are independent if the occurrence of one event does not affect the occurrence of any of the others. If one coin is selected at random from a boX containing dimes and quarters and is replaced, the result of this selection would not affect the result of a second selection. These two drawings are independent events.
The probability that two independent events will both happen is the product of the separate probabilities. If X and Y are independent events, the probability that both X and Y will happen may be found by the formula
P(X and Y) = P(X) X P(Y)
Example:
If a die is thrown twice, what is the probability that a five will come up on the first throw, and a three will come up on the second throw?
Solution:
The probability that a five will come up the first time is 1/6. The probability that a three will come up the second time is 1/6.
P(5 and 3) = P(5) x P(3) = 1/6 x 1/6 = 1/36
The probability that both events will happen is 1/36.
To determine the probability of event X or event Y happening, when the two events are not mutually exclusive:
1.) Find the sum of the separate probabilities.
2.) From this sum subtract the probability that both events will occur.
P(X or Y) = P(X) + P(Y)—P(X and Y)
Example 1:
When a card is drawn at random from a normal deck of 52 cards, what is the probability that it will be either an ace or a spade?
Solution:
The probability of drawing an ace from the deck is 4/52.
The probability of drawing a spade is 13/52. The probability of drawing the ace of spades is 1/52.
The probability of drawing either an ace or a spade is:
P(ace) + P(spade)—P(ace of spades)
4/52 + 13/52—1/52 = 16/52 = 4/13
Example 2:
If two dice are thrown, what is the probability that one of them will come up less than five?
Solution: The dice can come up the following ways:
11
|
21
|
31
|
41
|
51
|
61
|
12
|
22
|
32
|
42
|
52
|
62
|
13
|
23
|
33
|
43
|
53
|
63
|
14
|
24
|
34
|
44
|
54
|
64
|
15
|
25
|
35
|
45
|
55
|
65
|
16
|
26
|
36
|
46
|
56
|
66
|
The probability that the first die will come up less than 5 is 24/36.
The probability that the second die will come up less than 5 is 24/36.
The probability that both dice will come up less than 5 is 16/36.
The probability that one of them will come up less than 5 is
24/36 + 24/36—16/36 = 32/36 = 8/9
Exercises:
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1.) If the probabilities that Mary and Sue will receive awards in a contest are 3/5 and 1/3 respectively, what is the probability that one or the other will receive an award?
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2.) If five coins are tossed, what is the probability that all five coins will turn up heads?
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3.) Find the probability that a person will throw 4, 8, and 10 on the first, second, and third tosses of a pair of dice.
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4.) If two dice are thrown, what is the probability that one of them will come up greater than four?
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5.) A bag contains six white balls, four green balls, and three brown balls. If three balls are drawn, one at a time, and the ball is replaced after each drawing, what is the probability that the balls drawn will be green, white and brown?
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6.) A box contains four spools of black thread, six spools of brown thread, and ten spools of white thread. A spool is drawn, replaced, then a second spool is drawn. What is the probability that either a black or a brown spool is drawn?
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7.) A card is drawn from a standard deck of 52 cards, replaced, and a second card is drawn. What is the probability that both cards are tens?
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8.) The probability that Joe will solve a certain problem is 3/5, that Jane will solve it is 5/6, and that Sam will solve it is 1/4, What is the probability that Joe and Jane will solve it, and Sam will not solve it?