# 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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## Probability

P(E)= h/n P(not E) = f/n

The sum of the probability that an event will happen, and the probability that it will fail to happen is 1.

h/n + f/n = 1

If it is certain that an event will happen, the probability of a favorable outcome is 1. If an event is certain to fail, the probability that it will happen is 0. The probability that an event will happen, P(E), ranges from 0 to 1.

0 P(E) 1

Example 1: What is the probability of throwing a four in one throw of a die?

Solution: The die may fall in any one of six ways. Only one of these will be “4”. The probability of throwing a four is 1/6.

Example 2: The faces of a cube are marked with the letters A, A, B, C, D, E. If the cube is tossed, what is the probability that an A will turn up?

Solution: The cube may turn up six different ways. There are two ways in which an A can turn up. The probability that an A will turn up is 2/6 or 1/3.

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Example 3:
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A committee of three is to be chosen from ten girls. If Ann, Betty and Carol are among the group of ten girls, what is the probability that all three of them will be on the committee?

Solution: The total number of committees of three girls that can be chosen from ten girls is

10
^{
C
}
3 = 10! = 10 x 9 x 8 x 7! = 120

(10-3! 3! = 7! x 3 x 2

Ann, Betty, and Carol form one of these selections. The probability that the committee will consist of Ann, Betty and Carol is 1/120.

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Exercises
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- 1.) A bag contains 24 balls. Five of the balls are red, four are green, seven are blue, and eight are yellow. What is the probability that a ball picked at random will be (a) red? (b) green? (c) blue? (d) yellow?
- 2.) There are twenty-eight students in a class. Sixteen are girls, and twelve are boys. Find the probability that a student selected at random will be a girl.
- 3.) Find the probability that a number selected at random from the set of numbers 5, 6, 7, 10, 12, 14, 17, 21, 28, 30 will be divisible by 7.
- 4.) If you select a letter at random from the alphabet, what is the probability that it will be a consonant?
- 5.) If a number is selected at random from the set of numbers 1, 3, 17, 25, 71, what is the probability that the number is (a) an odd digit? (b) an even digit? (c) divisible by 3? (d) a prime number? (e) a composite number?
- 6.) If two dice are thrown, what is the probability of getting a sum of eight?
- 7.) Four marbles are drawn at random from a bag containing five orange marbles and seven brown marbles. What is the probability that (a) all four marbles are orange? (b) all four marbles are brown?
- 8.) If six cards are drawn at random from a deck of 52 cards, what is the probability that they are all spades?
- 9.) If a coin is thrown, what is the probability that it will turn up “tails”?
- 10.) In Hillcross High School there are 300 freshmen, 280 sophomores, 275 juniors, and 256 seniors. What is the probability that a student selected at random will be (a) a freshman? (b) a sophomore? (c) a junior? (d) a senior?