Elementary Statistics Triola, Elementary Statistics …ronharrow.com/wp-content/uploads/2012/08/Unit-10-The-Binomial... · Elementary Statistics Triola, Elementary Statistics 11/e

Elementary Statistics

Triola, Elementary Statistics 11/e

Unit 10 The Binomial Distribution

41 Copyright © RHarrow 21013 V130723

A commonly encountered discrete probability distribution is the binomial probability distribution. The following experiment will generate one:

We roll the dice 10 times and ask, what is the probability that we will get exactly four sevens? In order to be able to use the binomial distribution to answer this question, the following requirements have to be met.

1. The process has a fixed number of trials. In our case, we roll the dice ten time.

2. The trials must be independent. Each roll is independent of the other rolls; no roll depends on a previous roll regardless of what the spectators are telling you at the casino.

3. Each trial must have only one of two outcomes. In our example, either we get a seven or we don’t, hence there only two outcomes.

4. The probability of a win or a success remains the same throughout all the trials. In our case,

given the we defined a “success” as rolling a seven, the probability of a success is

for each roll

There are tables for finding the probabilities of different events for when we are working with a binomial distribution. However, we are going to use Excel. For our example, we will use the BINOM.DIST tool, found on the Statistics function menu, and fill it out as shown below.

Number_s is the number of successes you’re testing for, in this case 4. Trials is the total number of times you roll the dice. Probability_s is the probability of a success or in our case, rolling a seven on any one roll. Cumulative is set to false if we want exactly 4 times. As you can see in the middle of the window, the probability of rolling a 7 exactly four times out of ten is 0.0543 (rounded to four decimal places).





We can phrase probability questions in different ways. Below is a table of various different relationships that can be expressed, both in words and symbols. In addition, I have also shown the settings for the Cumulative input both for the Binomial Distribution and for later, when we work with the Poisson distribution.

Relationship Symbol Example Cumulative

Equals The probability of rolling exactly 5 sevens False

Less than The probability of rolling less than 5 sevens True

At most The probability of rolling at most 5 sevens True

Greater than The probability of rolling more than 5 sevens True

At least The probability of rolling at least 5 sevens True

Equals Synonyms: is, exactly This one is pretty straight forward. The following would be the input to BINOM.DIST,

Number_s: 5 Trials: Depends on the problem statement Probability: Depends on the problem statement Cumulative: False

Less than Synonyms: can’t think of any This one requires an explanation. First, the tool BINOM.DIST can only work with two

mathematical relationships, Furthermore, we are dealing with discrete numbers here. Less than 5 is the same as less than or equal to 4. THINK ABOUT THAT. Since we cannot use the tool for a strictly less than situation, we have to use which means, that instead of finding ( ) we find ( ) This is an important concept, so if you have to, reread this paragraph until you understand it.

Number_s: 4 Trials: Depends on the problem statement Probability: Depends on the problem statement Cumulative: True

At most Synonyms: At most means less than or equal so we can use the tool in a fairly straight forward manner. We are asking for ( ) Another synonym is “no more than”.

Number_s: 5 Trials: Depends on the problem statement Probability: Depends on the problem statement Cumulative: True





Here’s an example of showing how to find the probability of rolling no more than 4 sevens.

We now see that the probability of rolling a 7 no more than four times has jumped to 0.9845.

Finally, how would we find the probability of rolling a 7 at least four times. This means that we would include the cases of rolling a 7 four times, five times, six, seven, eight, nine, or ten times.

Take a moment here and think about the differences and similarities between, no more than four, at least four, four or less, four or more. Write out the mathematical symbol for each expression. For example, exactly four times is

1. At least four times: Answer #1 on the Answer Sheet.

2. No more than four times: Answer #2 on the Answer Sheet.

3. Four or more times: Answer #3 on the Answer Sheet.

4. At most four times: Answer #4 on the Answer Sheet.

Finally, how would we use the tool to find the probability of rolling a 7 at least four times? The problem that we encounter is that the tool can only directly give us answers for either “equals” or “less than or equal”. What do you think we should do?

Consider the set of all possibilities:

{ }

Each element is the number of times a seven can come up. What is the set consisting of at least four

times? Answer #5 on the Answer Sheet.

What is the complement of the set in #6 above? Answer #6 on the Answer Sheet.

What is the probability of getting the set in #6, i.e. the probability of rolling at most 3 sevens? Use the

stat tool to find the answer and write the answer for #7 on the Answer Sheet.





Using the rule that ( ̅) ( ) find the probability of getting 4 or more sevens. Answer #8 on

the Answer Sheet.

Here’s an interesting case study where the binomial distribution can be used. Kim is a recent graduate

in engineering from a good school. A prestigious engineering firm is looking to fill 20 positions with new

graduates. There are hundreds of applications, and the percentage of female applicants is 30%.

We’re going to make the assumption that all of the applicants were equally qualified. Therefore, if all

the applications were put in a large basket and one was randomly selected, what would be the

probability of selecting a female applicant? Answer #9 on the Answer Sheet.

Kim did not receive an offer, and later she found out that only two women out of twenty were hired.

Kim suspected that there had been bias against women in the hiring process. With your knowledge of

statistics, use the binomial distribution to tell Kim what would be the probability of randomly selecting

two or fewer women for the twenty open positions, i.e. selecting only two or fewer women by chance,

and furthermore, tell Kim whether or not it was an unusual event.

What is the number of Trials, i.e. how many chances did a women have to be hired? Answer #10 on the

Answer Sheet.

We are looking for the probability that two or fewer women were hired. Therefore, what should

Number_s be and what value are we going to use for Cumulative. Answer #11 and #12 on the Answer

Sheet.

What value should we use for Probability_s, i.e. the probability of randomly selecting a woman to be

hired? Answer #13 on the Answer Sheet.

Now use BINOM.DIST to find the probability that two or fewer women were hired given that there was

no bias, and report on whether you think that this is an unusual event, explaining why. Answer #14 and

#15 on the Answer Sheet.

In statistics, if, under a given assumption, the probability of an observed event is extremely small, we

conclude that the assumption is probably not correct. This is the Rare Event Rule. Any probability of

less than 5% is considered to be extremely small. The assumption here was that there was no bias in the

hiring process. What would you tell Kim? Answer #16 on the Answer Sheet.

Note: Whenever an exercise in MyMathLab says to use tables to look up probabilities, you

should use Excel instead.

This is the end of Unit 10. In class, you will get more practice with these

concepts by working exercises in MyMathLab.





____________________________

Name

Answer Sheet

1. ______________

2. ______________

3. ______________

4. ______________

5. ______________

6. ______________

7. ______________

8. ______________

9. ______________

10. ______________

11. ______________

12. ______________

13. ______________

14. ______________

15. ______________________________________________________________________________

______________________________________________________________________________

16. ______________________________________________________________________________

______________________________________________________________________________

Documents

Elementary Statistics Triola, Elementary Statistics …ronharrow.com/wp-content/uploads/2012/08/Unit-10-The-Binomial... · Elementary Statistics Triola, Elementary Statistics 11/e