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Ch. 3: Probability Topics

Module by: Susan Dean, Barbara Illowsky, Ph.D.. E-mail the authors

Summary: This module serves as the complementary teacher's guide for the Probability Topics chapter of the Elementary Statistics textbook/collection.

The best way to introduce the terms is through examples. You can introduce the terms experiment, outcome, sample space, event, probability, equally likely, conditional, mutually exclusive events, and independent events AND you can introduce the addition rule, the multiplication rule with the following example: In a box (you cannot see into it), there are are 4 red cards numbered 1, 2, 3, 4 and 9 green cards numbered 1, 2, 3, 4, 5, 6, 7, 8, 9. You randomly draw one card (experiment). Let RR be the event the card is red. Let GG be the event the card is green. Let EE be the event the card has an even number on it.

Example 1

Event Card Example

  1. List all possible outcomes (the sample space). Have students list the sample space in the form {R1, R2, R3, R4, G1, G2, G3, G4, G5, G6, G7, G8, G9}. Each outcome is equally likely. Plane outcome = 1 13 1 13 size 12{ { { size 8{1} } over { size 8{"13"} } } } {} .
  2. Find P ( R ) P ( R ) size 12{P \( R \) } {} .
  3. Find P ( G ) P ( G ) size 12{P \( G \) } {} . G is the complement of R. P ( G ) P ( G ) size 12{P \( G \) } {} + P ( R ) P ( R ) size 12{P \( R \) } {} = _______.
  4. P(P( size 12{P \( } {}red card given a that the card has an even number on it) = P ( R E ) P ( R E ) size 12{P \( R \lline E \) } {} .This is a conditional. Pick the red card out of the even cards. There are 6 even cards.
  5. Find P(R AND E)P(R size 12{P \( R} {} AND E) size 12{E \) } {}. (Multiplication Rule: P(R and E ) = P ( E R ) ( P ( R ) P(R and E ) = P ( E R ) ( P ( R ) size 12{E \) =P \( E \lline R \) \( P \( R \) \) } {} )
  6. P(R OR E)P(R size 12{P \( R} {}OR E) size 12{E \) } {}. (Addition Rule: P(R OR E)=P(E)+P(R)P(E AND R)P(R size 12{P \( R} {} OR E)=P(E)+P(R)P(E size 12{E \) =P \( E \) +P \( R \) - P \( E} {} AND R) size 12{R \) } {})
  7. Are the events R R size 12{R} {} and G G size 12{G} {} mutually exclusive? Why or why not?
  8. Are the events G G size 12{G} {} and E E size 12{E} {} independent? Why or why not?

Example 2

Problem 1

(Optional Topic) A Venn diagram is a tool that helps to simplify probability problems. Introduce a Venn diagram using an example. Example: Suppose 40% of the students at ABC College belong to a club and 50% of the student body work part time. Five percent of the student body works part time and belongs to a club.

Have the students work in groups to draw an appropriate Venn diagram after you have shown them what a Venn diagram basically looks like. The diagram should consist of a rectangle with two overlapping circles. One rectangle represents the students who belong to a club (40%) and the other circle represents those students who work part time (50%). The overlapping part are those students who belong to a club and who work part time (5%).

Find the following:

  1. P(student works part time but does not belong to a club)P(student works part time but does not belong to a club)
  2. P(student belongs to a club given that the student works part time)P(student belongs to a club given that the student works part time)
  3. P(student does not belong to a club)P(student does not belong to a club)
  4. P(works part time given that the student belongs to a club)P(works part time given that the student belongs to a club)
  5. P(student belongs to a club or the student works part time)P(student belongs to a club or the student works part time)

Solution

Figure 1
A venn diagram with a 5% overlap
  • C: student belongs to a club
  • PT: student works part time

Example 3

Problem 1

Find the following:

  1. P(a child is 9 - 11 years old)P(a child is 9 - 11 years old)
  2. P(a child prefers regular soccer camp)P(a child prefers regular soccer camp)
  3. P(a child is 9 - 11 years old and prefers regular soccer camp)P(a child is 9 - 11 years old and prefers regular soccer camp)
  4. P(a child is 9 - 11 years old or prefers regular soccer camp)P(a child is 9 - 11 years old or prefers regular soccer camp)
  5. P(a child is over 14 given that the child prefers micro soccer camp)P(a child is over 14 given that the child prefers micro soccer camp)
  6. P(a child prefers micro soccer camp given that the child is over 14) P(a child prefers micro soccer camp given that the child is over 14)

Tree Diagrams (Optional Topic)

A tree is another probability tool. Many probability problems are simplified by a tree diagram. To exemplify this, suppose you want to draw two cards, one at a time, without replacement from the box of 4 red cards and 9 green cards.

Figure 2: There are (13)(12) = 156 Possible Outcomes. (ex. R1R1, R1R2, R1G3, G3G4, etc.)
A Tree diagram that illustrates the possible outcomes of the above card experiment

Example 4

Problem 1

Find the following:

  1. P(RR)P(RR)
  2. P(RG or GR)P(RG or GR)
  3. P(at most one G in two draws)P(at most one G in two draws)
  4. P(G on the 2nd draw|R on the 1st draw)P(G on the 2nd draw|R on the 1st draw). The size of the sample space has been reduced to 12+36=48112+36=481.
  5. P(no R on the 1st draw)P(no R on the 1st draw)

Introduce contingency tables as another tool to calculate probabilities. Let's suppose an owner of a soccer camp for children keeps information concerning the type of soccer camp the children prefer and their ages. The data is for 572 children.

Table 1
Type of Soccer Camp Preference Under 6 6-8 9-11 12-14 Over 14 Row Total
Micro 42 76 46 25 10 199
Regular 8 68 92 105 100 373
Column Total 50 144 138 130 110 572

Assign Practice

Assign Practice 1 and Practice 2 in class. Have students work in groups.

Assign Lab

The Probability Lab is an excellent way to cement many of the ideas of probability. The lab is a group effort (3 - 4 students per group).

Assign Homework

Assign Homework. Suggested problems: 1 - 15 odds, 19, 20, 21, 23, 27, 28 - 30.

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