Suppose you roll a die many times and record the number of threes you obtain. Is it possible to predict ahead of time the average number of threes you will obtain? The answer is ‘‘Yes.’’ It is calledexpected valueor themean
of a binomial distribution. This mean can be found by using the formula mean ()¼npwherenis the number of times the experiment is repeated and
pis the probability of a success. The symbol for the mean is the Greek letter
(mu).
EXAMPLE: A die is tossed 180 times and the number of threes obtained is recorded. Find the mean or expected number of threes.
SOLUTION:
n¼180 and p¼1
6 since there is one chance in 6 to get a three on each roll.
¼np¼1801
6
¼30
Hence, one would expect on average 30 threes.
EXAMPLE:Twelve cards are selected from a deck and each card is replaced before the next one is drawn. Find the average number of diamonds.
SOLUTION:
In this case,n¼12 andp¼1352 or 14since there are 13 diamonds and a total of 52 cards. The mean is
¼np
¼121
4
¼3
Hence, on average, we would expect 3 diamonds in the 12 draws.
Statisticians are not only interested in the average of the outcomes of a probability experiment but also in how the results of a probability experiment vary from trial to trial. Suppose we roll a die 180 times and record the number of threes obtained. We know that we would expect to get about 30 threes. Now what if the experiment was repeated again and again? In this case, the number of threes obtained each time would not always be 30 but would vary about the mean of 30. For example, we might get 28 threes one time and 34 threes the next time, etc. How can this variability be explained? Statisticians use a measure called the standard deviation. When the standard deviation of a variable is large, the individual values of the variable are spread out from the mean of the distribution. When the standard deviation of a variable is small, the individual values of the variable are close to the mean. The formula for the standard deviation for a binomial distribution is standard deviation¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffinpð1pÞ. The symbol for the standard deviation is the Greek letter (sigma).
CHAPTER 7
The Binomial Distribution
EXAMPLE: A die is rolled 180 times. Find the standard deviation of the number of threes. SOLUTION: n¼180, p¼1 6, 1p¼1 1 6¼ 5 6 ¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffinpð1pÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1801 6 5 6 r ¼pffiffiffiffiffi25 ¼5
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< þ2
In the die example, we can expect most values will fall between 3025<most values<30þ25
3010<most values<30þ10 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:
n¼100, p¼0.80, 1p¼1¼0.80¼0.20
mean: ¼np ¼100ð0:80Þ ¼80 standard deviation: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffinpð1pÞ ¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi100ð0:8Þð0:2Þ ¼pffiffiffiffiffi16 ¼4 Approximate range of values:
2 <most values < þ2 ðÞ
802 ð4Þ <most values<80 þ2 ð4Þ
72<most values<88
Hence, most of his scores will be somewhere between 72 and 88.
Note: The concept of the standard deviation is much more complex than what is presented here. Additional information on the standard deviation will be presented in Chapter 9. More information on the standard deviation can also be found in all statistics textbooks.
PRACTICE
1. Twenty cards are selected from a deck of 52 cards. Each card is replaced before the next card is selected. Find the mean and standard deviation of the number of clubs selected.
2. A coin is tossed 1000 times. Find the mean and standard deviation of the number of heads that will occur.
3. A 50-question multiple choice exam is given. There are four choices for each question. Find the mean and standard deviation of the number of correct answers a student will get if he selects each answer at random.
4. A die is rolled 720 times. Find the mean, standard deviation, and approximate range of values for the number of threes obtained. 5. A factory manufactures microchips of which 4% are defective. Find
the average number of defective microchips in a lot of 500. Also, find the standard deviation and approximate range of values.