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This chapter covers principles of probability. After completing this chapter students should be able to: write sample spaces; determine whether two events are mutually exclusive; use the addition rule; calculate probabilities using tree diagrams and combinations; solve problems involving conditional probability; determine whether two events are independent.

Sample spaces and probability

In problems 1 - 6, write a sample space for the given experiment.

A die is rolled.

1,2,3,4,5,6 size 12{ left lbrace 1,2,3,4,5,6 right rbrace } {}
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A penny and a nickel are tossed.

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A die is rolled, and a coin is tossed.

1H , 2H , 3H , 4H , 5H , 6H , 1T , 2T , 3T , 4T , 5T , 6T size 12{ left lbrace 1H,2H,3H,4H,5H,6H,1T,2T,3T,4T,5T,6T right rbrace } {}
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Two dice are rolled.

1 2 3 4 5 6
1 (1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6)
2 (2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6)
3 (3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6)
4 (4, 1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6)
5 (5, 1) (5, 2) (5, 3) (5, 4) (5, 5) (5, 6)
6 (6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6)
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A jar contains four marbles numbered 1, 2, 3, and 4. Two marbles are drawn.

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In problems 7 - 12, a card is selected from a deck. Find the following probabilities.

P an ace size 12{P left ("an ace" right )} {}
4 / 52 size 12{4/"52"} {}
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P a red card size 12{P left ("a red card" right )} {}
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P a club size 12{P left ("a club" right )} {}
13 / 52 size 12{"13"/"52"} {}
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P a face card size 12{P left ("a face card" right )} {}
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P a jack or spade size 12{P left ("a jack or spade" right )} {}
16 / 52 size 12{"16"/"52"} {}
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P a jack and a spade size 12{P left ("a jack and a spade" right )} {}
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A jar contains 6 red, 7 white, and 7 blue marbles. If a marble is chosen at random, find the following probabilities.

P red size 12{P left ("red" right )} {}
6 / 20 size 12{6/"20"} {}
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P white size 12{P left ("white" right )} {}
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P red or blue size 12{P left ("red or blue" right )} {}
13 / 20 size 12{"13"/"20"} {}
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P red and blue size 12{P left ("red and blue" right )} {}
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Consider a family of three children. Find the following probabilities.

P two boys and a girl size 12{P left ("two boys and a girl" right )} {}
3 / 8 size 12{3/8} {}
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P at least one boy size 12{P left ("at least one boy" right )} {}
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P children of both sexes size 12{P left ("children of both sexes" right )} {}
6 / 8 size 12{6/8} {}
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P at most one girl size 12{P left ("at most one girl" right )} {}
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Two dice are rolled. Find the following probabilities.

P the sum of the dice is 5 size 12{P left ("the sum of the dice is 5" right )} {}
4 / 36 size 12{4/"36"} {}
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P the sum of the dice is 8 size 12{P left ("the sum of the dice is 8" right )} {}
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P the sum is 3 or 6 size 12{P left ("the sum is 3 or 6" right )} {}
7 / 36 size 12{7/"36"} {}
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P the sum is more than 10 size 12{P left ("the sum is more than 10" right )} {}
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A jar contains four marbles numbered 1, 2, 3, and 4. If two marbles are drawn, find the following probabilities.

P the sum of the number is 5 size 12{P left ("the sum of the number is 5" right )} {}
4 / 12 size 12{4/"12"} {}
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P the sum of the numbers is odd size 12{P left ("the sum of the numbers is odd" right )} {}
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P the sum of the numbers is 9 size 12{P left ("the sum of the numbers is 9" right )} {}

0

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P one of the numbers is 3 size 12{P left ("one of the numbers is 3" right )} {}
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Mutually exclusive events and the addition rule

Determine whether the following pair of events are mutually exclusive.

A = A person earns more than $25,000 size 12{A= left lbrace "A person earns more than $25,000" right rbrace } {}
B = A person earns less than $20,000 size 12{B= left lbrace "A person earns less than $20,000" right rbrace } {}

Yes

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A card is drawn from a deck.

C = It is a King size 12{C= left lbrace "It is a King" right rbrace } {} D = It is a heart size 12{D= left lbrace "It is a heart" right rbrace } {} .
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A die is rolled.

E = An even number shows size 12{E= left lbrace "An even number shows" right rbrace } {}
F = A number greater than 3 shows size 12{F= left lbrace "A number greater than 3 shows" right rbrace } {}

No

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Two dice are rolled.

G = The sum of dice is 8 size 12{G= left lbrace "The sum of dice is 8" right rbrace } {}
H = One die shows a 6 size 12{H= left lbrace "One die shows a 6" right rbrace } {}
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Three coins are tossed.

I = Two heads come up size 12{I= left lbrace "Two heads come up" right rbrace } {}
J = At least one tail comes up size 12{J= left lbrace "At least one tail comes up" right rbrace } {}

No

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A family has three children.

K = First born is a boy size 12{K= left lbrace "First born is a boy" right rbrace } {}
L = The family has children of both sexes size 12{L= left lbrace "The family has children of both sexes" right rbrace } {}
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Use the addition rule to find the following probabilities.

A card is drawn from a deck, and the events C size 12{C} {} and D size 12{D} {} are as follows:

C = It is a king size 12{C= left lbrace "It is a king" right rbrace } {}
D = It is a heart size 12{D= left lbrace "It is a heart" right rbrace } {}

Find P C   or   D size 12{P left (C" or "D right )} {} .

16 / 52 size 12{"16"/"52"} {}
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A die is rolled, and the events E size 12{E} {} and F size 12{F} {} are as follows:

E = An even number shows size 12{E= left lbrace "An even number shows" right rbrace } {}
F = A number greater than 3 shows size 12{F= left lbrace "A number greater than 3 shows" right rbrace } {}

Find P E   or   F size 12{P left (E" or "F right )} {} .

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Two dice are rolled, and the events G size 12{G} {} and H size 12{H} {} are as follows:

G = The sum of dice is 8 size 12{G= left lbrace "The sum of dice is 8" right rbrace } {}
H = Exactly one die shows a 6 size 12{H= left lbrace "Exactly one die shows a 6" right rbrace } {}

Find P G   or   H size 12{P left (G" or "H right )} {} .

13 / 36 size 12{"13"/"36"} {}
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Three coins are tossed, and the events I size 12{I} {} and J size 12{J} {} are as follows:

I = Two heads come up size 12{I= left lbrace "Two heads come up" right rbrace } {}
J = At least one tail comes up size 12{J= left lbrace "At least one tail comes up" right rbrace } {}

Find P I   or  J size 12{P left (I" or "J right )} {} .

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At De Anza college, 20% of the students take Finite Mathematics, 30% take Statistics and 10% take both. What percentage of the students take Finite Mathematics or Statistics?

40%

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Source:  OpenStax, Applied finite mathematics. OpenStax CNX. Jul 16, 2011 Download for free at http://cnx.org/content/col10613/1.5
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