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[/l