16.2 Probability of Events Occurring Together Color coded headings key: Yellow – Definitions; Green – Notation; Blue – Examples; Peach – Rules & How To; Purple - Practice

Definitions Conditional Probability Conditional probability is the probability of an event given that you know another event has occurred.

The probability that event A occurs, given that event B has occurred. Notation Notation Read As Means P(A|B) “P of A given B” The probability that event A occurs, given that event B has occurred.

Guided Practice Let’s find the following probabilities given I randomly choose a person in the classroom. P(A girl) P(A girl | A person wearing pink) P(A girl | plays softball) P(A girl | plays lacrosse) P(Plays lacrosse | A girl)

Guided Practice There is a jar of 3 blue marbles and 5 yellow marbles without replacement. Find: P(BY) (means blue, then yellow) P(YB) (means yellow, then blue) P(YY) P(BB) P(2nd is B|1st is Y) P(2nd is Y|1st is Y) P(2nd is Y|1st is B)

Conditional Probability The probability of event A occurring, given that event B has occurred is: P(A|B) = P(A and B) P(B) Note: P(B) must not equal 0. “Given” goes in the denominator!

Example 2. Draw a card at random from a standard deck. Event A is you draw a queen. Event B is you draw a red card. What is the probability that you draw a queen, given that the card is red?

General Multiplication Rule The probability of two events both occurring is the probability of the first times the probability of the second, given that the first has occurred: P(A and B) = P(A) ∙ P(B|A) as well as P(A and B) = P(B) ∙ P(A|B) This is the same equation as conditional probability, just solved for P(A and B).

Definitions Independent Events Two events are said to be independent if knowing the outcome of one event does not change the probability of the other event occurring. That is, P(B|A) = P(B) and P(A|B) = P(A)

Examples of Independence The probabilities of choosing a red card and queen from a standard deck of cards are independent. The probabilities of choosing a girl at random from the class and choosing a softball player are not independent.

Multiplication Rule for Independent Events The probability of two independent events both occurring is the product of their individual probabilities: P(A and B) = P(A) ∙ P(B) Note: If this multiplication rule holds, then two events are independent. Ex. Find the P(Heads and Rolling a 3) This is the same equation as the general multiplication rule, because P(B|A) = P(B)

Guided Practice You draw one card from a standard deck. Event A is the card is a heart. Event B is the card is face card. Are these events independent? How can you know? State an Event in this sample space that would NOT be independent from Event A. Yes, they are independent. 13 of the 52 cards are hearts. 12 of the 52 cards are face cards. 3 or the 52 cards are both. The two ways you can know are P(A)P(B) = P(A and B) , or (13/52)*(3/13) = 3/52 or, alternatively P(A) = (PA|B) , or 1/4= 3/12 Answers may vary. One is to choose a disjoint event, as disjoint events are NEVER independent! Choose Event C to be a black card is chosen. Then Event A and Event C are not independent, because if you know the card is a heart, then the probability of it also being black is zero. An example of a non-disjoint event that is not independent is more difficult in this scenario, because all the suits have the same proportionality, but here is one: Let Event D be the event that the card is a Heart OR a Club. Then P(D)=26/52, and P(A)=13/52, but P(A|D)=13/26. Events A and D are not disjoint, and not independent.