Probability Sample space A sample space is defined

Probability

Sample space • A sample space is defined as a universal set of all possible outcomes from a given experiment.

Finding sample space Experiment 2: A spinner has 4 equal sectors colored yellow, blue, green and red. What is the probability of landing on each color after spinning this spinner? Sample Space: {yellow, blue, green, red} Probabilities: P(yellow) = 1/4 P(blue) = P(green) = 1 4 = 1 4 P(red) 1 4

Union • The union of two sets is a new set that contains all of the elements that are in at least one of the two sets. The union is written as:

Intersection • The intersection of two sets is a new set that contains all of the elements that are in both sets. The intersection is written as:

Handout #1 • Notice the different types of unions and intersections within the sample space (S). This handout will become very important as we look further into probability. Keep this handout in your notes.

INDEPENDENT Events • When two events are said to be independent of each other, what this means is that the probability that one event occurs in no way affects the probability of the other event occurring. • An example of two independent events is as follows; say you rolled a die and flipped a coin. The probability of getting any number face on the die in no way influences the probability of getting a head or a tail on the coin.

Finding independent events Multiplication Rule 1: When two events, A and B, are independent, the probability of both occurring is: P(A and B) = P(A) · P(B) Handout 2 #1

Dependent Events • When two events are said to be dependent, the probability of one event occurring influences the likelihood of the other event. • For example, if you were to draw a two cards from a deck of 52 cards. If on your first draw you had an ace and you put that aside, the probability of drawing an ace on the second draw is greatly changed because you drew an ace the first time.

Conditional Probability • Conditional probability deals with looking at probability of an event given that some other event first occurs. • Conditional probability is denoted by the following: • The above is read as the probability that B occurs given that A has already occurred • The above is mathematically defined as: .

Finding Conditional Probability To find the probability of the two dependent events, we use a modified version of Multiplication Rule 1 Multiplication Rule 2: When two events, A and B, To find the probability of the two dependent events, we use a modified are dependent, the probability version of Multiplication Rule 1, which was presented in the last lesson. of both occurring is: P(A and B) = P(A) · P(B|A) Handout 2 #2

Conditional Probability Cont. The conditional probability of an event B in relationship to an event A is the probability that event B occurs given that event A has already occurred. The notation for conditional probability is P(B|A), read as the probability of B given A. The formula for conditional probability is: Handout #3

Mutually Exclusive Experiment 1: A single 6 -sided die is rolled. What is the probability of rolling a 2 or a 5? Possibilities: 1. The number rolled can be a 2. 2. The number rolled can be a 5. Events: These events are mutually exclusive since they cannot occur at the same time. Probabilities: How do we find the probabilities of these mutually exclusive events? We need a rule to guide us.

Non-mutually exclusive Experiment 4: A single card is chosen at random from a standard deck of 52 playing cards. What is the probability of choosing a king or a club? In Experiment 4, the events are non-mutually exclusive. The addition causes the king of clubs to be counted twice, so its probability must be subtracted. When two events are nonmutually exclusive, a different addition rule must be used. 4 52 = 16 52 = 4 13 + 13 52 - 1 52

Addition Rule 1: Addition Rule When two events, A and B, are mutually exclusive, the probability that A or B will occur is the sum of the probability of each event. P(A or B) = P(A) + P(B) Addition Rule 2: When two events, A and B, are non-mutually exclusive, the probability that A or B will occur is: P(A or B) = P(A) + P(B) - P(A and B) Addition Rule 1& 2 handout