Chapter 3: Probability Topics
3.3 Two Basic Rules of Probability
Learning Objectives
By the end of this section, you should be able to:
- state and apply the multiplication rule to any two events
- state and apply the multiplication rule for independent events
- state and apply the addition rule for any two events
- state and apply the addition rule for mutually exclusive events
Recall the different combinations of relationships between two events:
| Independent? | |||
| Yes | No | ||
| Disjoint? | Yes | 1* | 2 |
| No | 3 | 4 | |
We must always go into a problem assuming two events are not mutually exclusive or independent. This “default” starting point is illustrated by the 4th position in the table above. Depending on the information you are given and assumptions you are able to make, you may move potions on this grid. Where you fall on the grid will dictate how we apply the rules we will discuss in this section to find probabilities of compound events.
There are two types of compound events we may be interested in, Unions (OR) and Intersections (AND), each with their own set of rules and assumptions. When calculating probability, there are two rules to consider when determining if the two events are independent or dependent and if they are mutually exclusive or not.
*Note: You will rarely, if ever, find yourself in this case
The Multiplication Rule
If [latex]A[/latex] and [latex]B[/latex] are two events defined on a sample space, then: [latex]P(A \text{ AND } B) = P(B) P(A|B)[/latex], or equivalently, [latex]P(A \cap B) = P(B) P(A | B)[/latex].
This rule may also be written as: [latex]P(A|B) = \frac{P\left(A\text{ AND }B\right)}{P\left(B\right)},[/latex] that is, the probability of [latex]A[/latex] given [latex]B[/latex] equals the probability of both [latex]A[/latex] and [latex]B[/latex] divided by the probability of [latex]B[/latex].
If [latex]A[/latex] and [latex]B[/latex] are independent, then the probability of A does not depend on the probability of B and [latex]P(A | B ) = P(A)[/latex]. As a result, [latex]P(A \text{ AND }B) = P(A|B) P(B)[/latex] becomes [latex]P(A \text{ AND }B) = P(A) P(B)[/latex].
Example
A school has 200 seniors of whom 140 will be going to college next year. Forty will be going directly to work. The remainder are taking a gap year. Of the seniors going to college, fifty play sports. Of the seniors going directly to work, thirty play sports. Five of the seniors taking a gap year play sports.
- What is the probability that a senior is taking a gap year?
- Are “taking a gap year” and “playing sports” independent events?
- Are “going to college” and “taking a gap year” independent events?
Your Turn!
Suppose you are going to roll a dice twice. Show that A = “Roll a 6 on the first roll” and B = “Roll a 4 on the second roll” are independent events using the Multiplication Rule.
The Addition Rule
If [latex]A[/latex] and [latex]B[/latex] are defined on a sample space, then: P(A OR B) = P(A) + P(B) – P(A AND B).
If [latex]A[/latex] and [latex]B[/latex] are mutually exclusive, then P(A AND B) = 0. Then P(A OR B) = P(A) + P(B) – P(A AND B) becomes P(A OR B) = P(A) + P(B).
Example
Klaus is trying to choose where to go on vacation. His two choices are: A = New Zealand and B = Alaska. Klaus can only afford one vacation; he is leaning towards New Zealand so the probability that he chooses A is P(A) = 0.6 and the probability that he chooses B is P(B) = 0.35.
- P(A AND B) = 0 because Klaus can only afford to take one vacation.
- Therefore, the probability that he chooses either New Zealand or Alaska is P(A OR B) = P(A) + P(B) = 0.6 + 0.35 = 0.95. Note that the probability that he does not choose to go anywhere on vacation must be 0.05.
Example
A student goes to the library. Let events B = the student checks out a book and M = the student checks out a movie. Suppose that P(B) = 0.40, P(M) = 0.30 and [latex]P(M|B) = 0.5[/latex].
- Find P(B AND M).
- Find P(B OR M).
Your Turn!
Carlos plays college soccer. He makes a goal 65% of the time he shoots. Carlos is going to attempt two goals in a row in the next game. A = the event Carlos is successful on his first attempt. P(A) = 0.65. B = the event Carlos is successful on his second attempt. P(B) = 0.65. Carlos tends to shoot in streaks. The probability that he makes the second goal GIVEN that he made the first goal is 0.90.
a. What is the probability that he makes both goals?
b. Are A and B independent?
c. What is the probability that Carlos makes either the first goal or the second goal?
d. Are A and B mutually exclusive?
Your Turn!
A community swim team has 150 members. Seventy-five of the members are advanced swimmers. Forty-seven of the members are intermediate swimmers. The remainder are novice swimmers. Forty of the advanced swimmers practice four times a week. Thirty of the intermediate swimmers practice four times a week. Ten of the novice swimmers practice four times a week. Suppose one member of the swim team is chosen randomly.
- What is the probability that the member is a novice swimmer?
- What is the probability that the member practices four times a week?
- What is the probability that the member is an advanced swimmer and practices four times a week?
- What is the probability that a member is an advanced swimmer and an intermediate swimmer? Are being an advanced swimmer and an intermediate swimmer mutually exclusive? Why or why not?
- Are being a novice swimmer and practicing four times a week independent events? Why or why not?
Let us now look at some examples where we include the complement of events. As a reminder, the complement of an event [latex]A[/latex] is denoted [latex]A'[/latex], and [latex]P(A') = 1-P(A)[/latex].
Example
Studies show that about one woman in seven (approximately 14.3%) who live to be 90 will develop breast cancer. Suppose that of those women who develop breast cancer, a test is negative 2% of the time. Also suppose that in the general population of women, the test for breast cancer is negative about 85% of the time. Let B = woman develops breast cancer and let N = tests negative. Suppose one woman is selected at random.
- What is the probability that the woman develops breast cancer? What is the probability that the woman tests negative?
- Given that the woman has breast cancer, what is the probability that she tests negative?
- What is the probability that the woman has breast cancer and tests negative?
- What is the probability that the woman has breast cancer or tests negative?
- Are having breast cancer and testing negative mutually exclusive?
- Given that a woman develops breast cancer, what is the probability that she tests positive?
- What is the probability that a woman develops breast cancer and tests positive?
- What is the probability that a woman does not develop breast cancer?
- What is the probability that a woman tests positive for breast cancer?
Your Turn!
Returning to the example, a student goes to the library, where events B = the student checks out a book and M = the student checks out a movie. Suppose that P(B) = 0.40, P(M) = 0.30 and P(M|B) = 0.5.
- Find P(B′ ).
- Find P(M AND B).
- Find P(B | M).
- Find P(M AND B′).
- Find P(M | B′ ).
References
DiCamillo, Mark, Mervin Field. “The File Poll.” Field Research Corporation. Available online at http://www.field.com/fieldpollonline/subscribers/Rls2443.pdf (accessed May 2, 2013).
Rider, David, “Ford support plummeting, poll suggests,” The Star, September 14, 2011. Available online at http://www.thestar.com/news/gta/2011/09/14/ford_support_plummeting_poll_suggests.html (accessed May 2, 2013).
“Mayor’s Approval Down.” News Release by Forum Research Inc. Available online at http://www.forumresearch.com/forms/News Archives/News Releases/74209_TO_Issues_-_Mayoral_Approval_%28Forum_Research%29%2820130320%29.pdf (accessed May 2, 2013).
“Roulette.” Wikipedia. Available online at http://en.wikipedia.org/wiki/Roulette (accessed May 2, 2013).
Shin, Hyon B., Robert A. Kominski. “Language Use in the United States: 2007.” United States Census Bureau. Available online at http://www.census.gov/hhes/socdemo/language/data/acs/ACS-12.pdf (accessed May 2, 2013).
Data from the Baseball-Almanac, 2013. Available online at www.baseball-almanac.com (accessed May 2, 2013).
Data from U.S. Census Bureau.
Data from the Wall Street Journal.
Data from The Roper Center: Public Opinion Archives at the University of Connecticut. Available online at http://www.ropercenter.uconn.edu/ (accessed May 2, 2013).
Data from Field Research Corporation. Available online at www.field.com/fieldpollonline (accessed May 2,2 013).
Two events that cannot happen at the same; they share no common outcomes
The occurrence of one event has no effect on the probability of the occurrence of another event
The occurrence of one event has no effect on the probability of the occurrence of another event. Events A and B are independent if one of the following is true:
• P(A|B) = P(A)
• P(B|A) = P(B)
• P(A AND B) = P(A)P(B)
Two events are mutually exclusive if the probability that they both happen at the same time is zero. If events A and B are mutually exclusive, then P(A AND B) = 0.
