Page 130 - Probability Demystified

CHAPTER 7 The Binomial Distribution 119

EXAMPLE: An archer hits the bull’s eye 80% of the time. If he shoots
5 arrows, find the probability that he will get 4 bull’s eyes.

SOLUTION:

n ¼ 5, x ¼ 4, p ¼ 0.8, 1 p ¼ 1 0.8 ¼ 0.2

4 1
Pð4 bull’s eyesÞ¼ C ð0:8Þ ð0:2Þ
5
4
¼ 5 0:08192
¼ 0:4096
In order to construct a probability distribution, the following formula
is used:
x
n C x p (1 p) n x where x ¼ 1, 2, 3, .. . n.
The next example shows how to use the formula.

EXAMPLE: A die is rolled 3 times. Construct a probability distribution for
the number of fives that will occur.

SOLUTION:
In this case, the die is tossed 3 times, so n ¼ 3. The probability of getting a 5
1
on a die is , and one can get x ¼ 0, 1, 2, or 3 fives.
6
3
0
1 5
For x ¼ 0, C 0 ¼ 0:5787
3
6
6
2
1
1 5
For x ¼ 1, C 1 ¼ 0:3472
3
6 6
1
2
1 5
For x ¼ 2, C 2 6 6 ¼ 0:0694
3
0
3
1 5
For x ¼ 3, C 3 ¼ 0:0046
3
6 6
Hence, the probability distribution is
Number of fives, x 0 1 2 3
Probability, P(x) 0.5787 0.3472 0.0694 0.0046