Chapter 4: Probability

Probability of an Event

Definitions An experiment is the process of observing a phenomenon that has variation in its outcomes. The sample space associated with an experiment is the collection of all possible distinct outcomes of the experiment. An elementary outcome is each outcome of the experiment. An event is the set of elementary outcomes possessing a designated feature

Examples m m

Examples Experiment: Rolling a pair of dice, dice 1 and dice 2. Event A: Both numbers are even: {(2,2),(4,4),(6,6)}. Event B: The second number is bigger than 4: {(1,5),(2,5), (3,5),(4,5), (5,5),(6,5), (1,6),(2,6), (3,6),(4,6), (5,6),(6,6)}

Examples Experiment: Tossing a coin twice. Sample space: {HH, HT, TH, TT)} (The first letter presents the outcome of the first toss, the second letter presents the outcome of the second toss) Event A: No tail: {HH, HT, TH} Event B: Head on the first toss: {HT, HH}

Three types of Sample Space Finite: sample spaces that have a finite number of elements. Countably infinite: sample spaces that have a countably infinite number of elements. Uncauntably infinite (Continuous): sample spaces that have a uncountably infinite number of elements.

Countably Infinite Sample Space Experiment: Suppose a gambler at a casino will continue pulling the handle of a slot machine until he hits the first jackpot. Observe the numbers of time he pulls the handle until he wins. Sample space: {1,2,3,4,5,6…..}

Uncountably Infinite Sample Space Experiment: Suppose a car with a full tank of gasoline is driven until its fuel runs out and the distance traveled, d, is recorded. Sample space: {All nonnegative numbers}

Probability of an Event Consider a finite sample space. The probability of an event is a numerical value that represents the proportion of times the event is expected to occur when the experiment is repeated many times under identical conditions. The probability of event A is denoted by P(A).

Probability of an Event