Page 134 - Probability Demystified

CHAPTER 7 The Binomial Distribution 123

EXAMPLE: A die is rolled 180 times. Find the standard deviation of the
number of threes.

SOLUTION:

1 1 5
n ¼ 180, p ¼ ,1 p ¼ 1 ¼
6 6 6
p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
¼ npð1 pÞ
r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 5
¼ 180
6 6
p ffiffiffiffiffi
¼ 25
¼ 5

The standard deviation is 5.
Now what does this tell us?

Roughly speaking, most of the values fall within two standard deviations
of the mean.

2 < most values < þ 2
In the die example, we can expect most values will fall between
30 2 5 < most values < 30 þ 2 5
30 10 < most values < 30 þ 10
20 < most values < 40

In this case, if we did the experiment many times we would expect between
20 and 40 threes most of the time. This is an approximate ‘‘range of values.’’
Suppose we rolled a die 180 times and we got only 5 threes, what can be
said? It can be said that this is an unusually small number of threes. It can
happen by chance, but not very often. We might want to consider some
other possibilities. Perhaps the die is loaded or perhaps the die has been
manipulated by the person rolling it!

EXAMPLE: An archer hits the bull’s eye 80% of the time. If he shoots 100
arrows, find the mean and standard deviation of the number of bull’s eyes.
If he travels to many tournaments, find the approximate range of values.

SOLUTION:

n ¼ 100, p ¼ 0.80, 1 p ¼ 1 ¼ 0.80 ¼ 0.20