Chapter 3: Probability Topics
Chapter 3 Review
Section 3.1 Review
In this module we learned the basic terminology of probability. The set of all possible outcomes of an experiment is called the sample space. Events are subsets of the sample space, and they are assigned a probability that is a number between zero and one, inclusive.
Formula Review
[latex]A[/latex] and [latex]B[/latex] represent events
[latex]0 \leq P(A) \leq 1[/latex]
If [latex]S[/latex] is the sample space of an experiment, [latex]P(S) = 1[/latex]
[latex]P(A | B) = \frac{P(A \text{ AND }B)}{P(B)}[/latex]
or equivalently, [latex]P(A | B) = \frac{P(A \cap B)}{P(B)}[/latex]
Section 3.2 Review
Two events [latex]A[/latex] and [latex]B[/latex] are independent if the knowledge that one occurred does not affect the chance the other occurs. If two events are not independent, then we say that they are dependent.
In sampling with replacement, each member of a population is replaced after it is picked, so that member has the possibility of being chosen more than once, and the events are considered to be independent. In sampling without replacement, each member of a population may be chosen only once, and the events are considered not to be independent. When events do not share outcomes, they are mutually exclusive of each other.
Formula Review
If [latex]A[/latex] and [latex]B[/latex] are independent, [latex]P(A \text{ AND } B) = P(A)P(B), P(A | B) = P(A)[/latex] and [latex]P(B | A ) = P(B)[/latex]. Equivalently, [latex]P(A \cap B) = P(A)P(B)[/latex].
If [latex]A[/latex] and [latex]B[/latex] are mutually exclusive, [latex]P(A \text{ OR } B)= P(A) + P(B)[/latex] and [latex]P(A \text{ AND }B) = 0[/latex]. Equivalently, [latex]P(A \cup B) = P(A) + P(B)[/latex] and [latex]P(A \cap B) = 0[/latex] because [latex]A \cap B = \emptyset[/latex].
Section 3.3 Review
The multiplication rule and the addition rule are used for computing the probability of [latex]A[/latex] and [latex]B[/latex], as well as the probability of [latex]A[/latex] and [latex]B[/latex] for two given events [latex]A[/latex] and [latex]B[/latex] defined on the sample space. In sampling with replacement each member of a population is replaced after it is picked, so that member has the possibility of being chosen more than once, and the events are considered to be independent. In sampling without replacement, each member of a population may be chosen only once, and the events are considered to be not independent. The events [latex]A[/latex] and [latex]B[/latex] are mutually exclusive events when they do not have any outcomes in common.
Formula Review
The multiplication rule: [latex]P(A \text{ AND } B) = P(A | B)P(B)[/latex]
or equivalently, [latex]P(A \cap B ) = P(A | B)P(B)[/latex]
The addition rule: [latex]P(A \text{ OR } B) = P(A) + P(B) - P(A \text{ AND } B)[/latex]
or equivalently, [latex]P(A \cup B) = P(A) + P(B) - P(A \cap B)[/latex]
Section 3.4 Review
There are several tools you can use to help organize and sort data when calculating probabilities. Contingency tables help display data and are particularly useful when calculating probabilities that have multiple dependent variables.
Section 3.5 Review
A tree diagram uses branches to show the different outcomes of experiments and makes complex probability questions easy to visualize.
A Venn diagram is a picture that represents the outcomes of an experiment. It generally consists of a box that represents the sample space S together with circles or ovals. The circles or ovals represent events. A Venn diagram is especially helpful for visualizing the OR event, the AND event, and the complement of an event and for understanding conditional probabilities.