Introduction to Probability and Probability Models

Probability of an Event A measure of the likelihood that an event will occur. Example: What is the probability of selecting a heart from a standard deck of cards?

Independent Events Two events, A and B, are independent if the fact that A occurs does not affect the probability of B occurring. When two events, A and B, are independent, the probability of both occurring is: Ex: What is the probability of selecting an ace from a standard deck and rolling a 3 on a standard 6-sided die? Selecting a card does not affect rolling a die. These events are independent. “AND” = Multiplication Rule

Mutually Exclusive Events Two events, A and B, are mutually exclusive if they can not occur at the same time. In other words: When two events, A and B, are mutually exclusive, the probability of either occurring is: Ex: If you select one card from a standard deck, what is the probability of selecting an ace or selecting the king of hearts? Notice that selecting an ace AND the king of hearts is impossible if you select one card. These events are mutually exclusive. “OR” = Division Rule

The Complement of an Event The complement of an event A, typically written Ā, is the set of all outcomes that are not A. The probability of an event and its complement always add up to 1: Ex: When tossing a standard 6-sided die, what is the probability of not getting a 5? The event of getting a 5 and the event of not getting a 5 are complements. The sum of their probabilities is 1.

Example Your teacher challenges you to a spinner game. You spin the two spinners with the probabilities listed below. The first letter should come from Spinner #1 and the second letter from Spinner #2. Find all of the possibilities and the probabilities of each possibility. T F I A U One way of finding an answer is listing the outcomes.

Example: Listing the Outcomes OUTCOMEPROBABILITY Another way of finding an answer is to use an Area Diagram. IT UT AT IF UF AF List all of the possibilities of the two spins The spins are independent. So we can multiply the probabilities. From the last slide, remember: P(I)=1/2 P(U)=1/6 P(A)=1/3 P(T)=1/4 P(F)=3/4 Spin an”I”, then a “T” P(I)xP(T) =

Example: Area Diagram Spinner #1 Spinner #2 ITUTAT IFUFAF IUA T F Reading the Diagram, the probability of rolling a “U” and a “T” is: Another way of finding an answer is to use a Tree Diagram.

Example: Tree Diagram STARTSTART I U A T F T F T F U T Reading the Diagram, the probability of rolling a “U” and a “T” is: