# Human Nature, Biology, and Social Structure: A Critical Look at WhatScience Can Tell Us About Society

## An Introduction to Mathematical Probability

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## 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)

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Example:
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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?

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Solution:
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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)

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Example 1:
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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?

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Solution:
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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

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Example 2:
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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 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

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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?
- 2.) If five coins are tossed, what is the probability that all five coins will turn up heads?
- 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.
- 4.) If two dice are thrown, what is the probability that one of them will come up greater than four?
- 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?
- 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?
- 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?
- 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?