1.Addition Rule 2.Multiplication Rule 3.Compliments 4.Conditional Probability 5.Permutation 6.Combinations 7.Expected value 8.Geometric Probabilities 9.Binomial Probabilities

Addition Rule for Non Mutually Exclusive Events 2 (A or B) = P(A) + P(B) – P(A and B) One card is drawn from a standard deck of cards. What is the probability that it is red or an ace? = P(Red) + P(Ace) – P(Both Red and Ace) = 26/52 + 4/52 – 2/52 = 28/52

Multiplication Rule Finding the probability of more than one event. The word “AND” is always used when describing the situation. 1)P(rolling a 4 and then a 2) = 1/6 *1/6 = 2.8% 2) P(rolling 3 odd #’s) = 3/6*3/6*3/6 = 12.5% 1)P(rolling a 4 and then a 2) = 1/6 *1/6 = 2.8% 2) P(rolling 3 odd #’s) = 3/6*3/6*3/6 = 12.5%

4 Example 1 continued  P(A1 AND A2) = P(A1)P(A2|A1) P(A1) = 4/52  There are now 3 aces left in a 51-card pack P(A2|A1) = 3/51  Overall: P(A1 AND A2) = (4/52) (3/51) =.0045 What’s the probability of pulling out two aces in a row from a deck of 52 cards?

If A is an event within the sample space S of an activity or experiment, the complement of A (denoted A') consists of all outcomes in S that are not in A. The complement of A is everything else in the problem that is NOT in A. Compliment: P(A') = 1 - P(A)

Conditional Probability and measures the probability of an event given that another event has occurred

1% of the population has disease X. If someone has the disease and gets tested the test is positive every time. If a healthy person gets tested for disease X they will get a false positive 10% of the time. If the lab comes back positive what will be the probability the person actually has the disease?

n = total number of items r = number chosen an arrangement of items in a particular order. Permutations

Permutations Examples A combination lock will open when the right choice of three numbers (from 1 to 30) is selected. How many different lock combinations are possible assuming no number is repeated?

Combinations an arrangement of items in which order does not matter. There are always fewer combinations than permutations. n = total number of items r = number chosen

Combinations Example To play a particular card game, each player is dealt five cards from a standard deck of 52 cards. How many different hands are possible?

a weighted average of all possible values where the weights are the probabilities of each outcome Expected value

Example: expected value probability distribution of ER arrivals x is the number of arrivals in one hour X P(x).4.2.1

Geometric Distribution want to find the number of trials for the 1 st success p = probability of success q = 1 – p = probability of failure X = # of trials until first success occurs p(x) = q x-1 p p = probability of success q = 1 – p = probability of failure X = # of trials until first success occurs p(x) = q x-1 p

15 Two Ways to use the Geometric Model #1: the probability of getting your first success on the x trail p(x) = q x-1 p #2: the number of trials until the first success is certain p(x) =

The desired probability is: p(x) = q x-1 p EXAMPLE: On Friday’s 25% of the customers at an ATM make deposits. What is the probability that it takes 4 customers at the ATM before the first one makes a deposit. ✔ Two Categories: Success: make a deposit Failure: don’t make a deposit ✔ Probability success same for each trial ✔ Wish to find the probability of the first EXAMPLE: On Friday’s 25% of the customers at an ATM make deposits. What is the probability that it takes 4 customers at the ATM before the first one makes a deposit. ✔ Two Categories: Success: make a deposit Failure: don’t make a deposit ✔ Probability success same for each trial ✔ Wish to find the probability of the first

n = number of trials x = number of successes n – x = number of failures p = probability of success in one trial q = 1 – p = probability of failure in one trial n = number of trials x = number of successes n – x = number of failures p = probability of success in one trial q = 1 – p = probability of failure in one trial BINOMIAL PROBABILITY finding the probability of a specific number of successes

EXAMPLE 2 You are taking a 10 question multiple choice test. If each question has four choices and you guess on each question, what is the probability of getting exactly 7 questions correct? p = 0.25 = guessing the correct answer q = 0.75 = guessing the wrong answer n = 10 x = 7

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