Learning Resources
Lesson
Investigation 15
Read the introduction and make sure you understand the problem that is presented. You have to determine if there is a bias towards males or did it occur by chance that four males and two females were hired when only 40% of the applicants were male.
This investigation will be best done as a small group calculating the required values and pooling results for the whole class.
In Step A, you have to design a simulation to model the problem. Since the problem specifies a 40% male population, flipping coins which simulates a 50% population cannot be used. You could construct a spinner that has one colour for 
(i.e. 60 out of 100) of the spinner to represent the females and 
of the spinner to represent the males. However, you may be best served using the random-number generator on your TI-83 (see Section01, lesson 02 if you have forgotten how to enter this into the calculator) or a spreadsheet if you are familiar with that software. Numbers from 1 to 40 inclusive will represent a male hired and numbers from 41 to 100 inclusive will represent a female being hired.
Six people are hired. Therefore, six successive numbers must be looked at as the number of males and females in each case. The simulation has to be repeated a sufficient number of times to generate a sampling distribution of the number in each case. Share the work, pool your results - each student complete a share such that at least 100 trials are recorded. Follow the interactive window below to complete this.
Once the trials are complete, summarize your results in a frequency table as below and record the experimental probability by dividing the results in each cell by the total.
Number of Males hired |
Frequency |
Experimental Probability |
0 |
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1 |
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2 |
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3 |
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4 |
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5 |
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6 |
For Step B, using your frequency table, construct your graph. Check the window below for a sample.
From the above sample you will note that the P(4 males) is approximately 0.2; with pooled results this probability will likely be much smaller. An exact answer will not be found, but based on the total number of pooled trials you can reasonably expect the probability to be very small. This can be verified from your probability bar graph.
Focus J
To continue the problem from the investigation we want to analyze it from a theoretically point of view using the Multiplication Principle and combinations. Read and study the explanation given in your textbook under the Focus (p.347).
The reasoning for the theoretical probability of choosing four males goes as follows:
- The probability of hiring a males is 0.4 (40%); based on the Multiplication Principle then the probability of hiring 4 males would be
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The hiring of 4 males is better described as hiring 4 males and 2 females since six people are hired. You must take into account the probability of hiring 2 females. The probability of hiring a female is 0.6(60%), so the probability of hiring 2 females is
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There are many different ways to chose four males from a group of six.. This number of ways is given by 6C4 .
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The overall probability is therefore given by
Compare this theoretical value to the experimental value found in the investigation. If you pooled your results and had a large number of trials to draw from it should be approaching this theoretical value. (Recall, for the 100 trials, we had 0.2 as an experimental probability.)
You should note that this is a binomial probability since it involves two items: choosing a male and a female. As well, it is important to keep in mind that 
; the complement of choosing a male. Also, the exponent for P(female) is the difference between the number of males and the size of the sample.
Before beginning the focus questions consider another example.
Example
Jeff can make a free throw 80% of the time. What is the probability that he will make 18 out of 20 baskets in a free-throw contest?
Solution
The probability of making a basket on any throw is 0.8; the probability of making 18 baskets is (0.8)18 .
Making 18 out of 20 baskets means that 2 baskets are missed or
The probability of "not making" 2 baskets is (0.2)2 .
For 20 throws, 18 baskets can be made in 20C18 ways. Therefore, the overall probability of making 18 baskets will be
Note this entire calculation can be done on your calculator; if you are not sure how to compete it check with your teacher.
You should now be ready to complete the assigned work on this lesson.
Activity
Investigation Questions p.346. Complete 1, 2 and 3
Focus Questions p.347. Complete 4, 5, 6 and 7
Check Your Understanding p.347 - 348. Complete 8, 9, 10 and 11
When you have completed these questions, ask your on-site teacher to get the solutions for you from the Teacher's Resource Binder and check them against your answers. After you do this, if there is something you had trouble with and still do not understand, contact your on-line teacher for help.
Test Yourself
Debbie builds model rockets. She has found that the model rocket engines misfire 5% of the time. She has built a new model for a competition and is allowed to bring only 12 engines. The rocket has to make 10 flights. What is the probability that she will lose three engines and not have enough engines to complete the competition?
Click here for a suggested solution.