The probability that it rains is 70% The probability that it does NOT rain is 30% Instinct tells us that for any event E, the probability that E happens + the probability that E does NOT happen = 100% P(E) + P(E’) = 100% = 100/100 = 1

Flip 3 coins: 2 x 2 x 2 = 8 possible outcomes H T H T H T H T H T H T H T

Flip 3 coins: 2 x 2 x 2 = 8 possible outcomes H T H T H T H T H T H T H T The probability that all land tails up = 1/8 The probability that at least one lands heads up = 7/8 ALL TAILS NOT ALL TAILS

FLIP 10 COINS What is the probability that at least one lands heads up? 10 The number of possible outcomes = 2 = 1024 P( all tails ) = 1/1024 P( NOT all tails ) = P(at least one heads up ) = 1023/1024

Roll 4 dice What is the probability that at least two land on the same number? Some of the possible outcomes: Favorable outcomes 6 x 6 x 6 x 6 = 1296 possible outcomes

Roll 4 dice It would be very difficult to count the number of ways in which at least two land on the same number. Instead let’s try to count the number of ways in which it does NOT happen that at least two are the same ….. That is They ALL land on DIFFERENT numbers # ways blue die can land: 6 # ways red die can land: x 5 # ways yellow die can land: x 4 # ways green die can land: x 3 = 360

Roll 4 dice # ways They ALL land on DIFFERENT numbers # ways blue die can land: 6 # ways red die can land: x 5 # ways yellow die can land: x 4 # ways green die can land: x 3 = x 6 x 6 x 6 = 1296 possible outcomes P( all different ) = 360/1296 P( NOT all different ) = /1296

1296/ / /1296

The probability that you get an A in French is 1/10 The probability that you get an B in French is 3/10 What is the probability that you get an A OR a B in French? Instinct tells us to add when we see the word OR 1/10 + 3/10 = 4/10 = P( A OR B ) In some cases, there is more to consider.

shuffle this deck of 10 cards and draw one at random. What is the probability that it is a white card OR a King ?

P ( white card ) = P( W ) = 5/10P ( King ) = P( K ) = 2/10 P ( white OR King ) = 6/10

P ( white card ) = P( W ) = 5/10P ( King ) = P( K ) = 2/10 P ( white OR King ) = 6/10

Draw 2 cards without replacement from a standard deck of 52 cards. What is the probability that both are red OR both are kings? # possible outcomes = # ways of drawing 2 red cards = # ways of drawing 2 kings = P ( both red ) = P( R ) = 325/1326 P ( both kings ) = P( K ) = 6/1326

P ( both red ) = P( R ) = 325/1326 P ( both kings ) = P( K ) = 6/1326 Both are red AND both are kings = only 1 of the 1326 possible hands satisfies this condition = 1/1326

P ( both red ) = P( R ) = 325/1326 P ( both kings ) = P( K ) = 6/1326 = 1/1326

There are 10 women in a study.

3 of them smoke S

There are 10 women in a study. 3 of them smoke S 3 develop heart disease H

There are 10 women in a study. S 3 develop heart disease H The probability that a woman in this sample develops heart disease = P(H)= 3/10

H S The probability that she gets heart disease IF she smokes =

S The probability that she gets heart disease IF she smokes = 2/3

H S P(H) = 3/10 = 30% P(H IF S) P(H/S) = 2/3 = 66.7% Based on this data, we might conclude that a woman is more than twice as likely to develop heart disease IF she smokes.

H S P(H IF S) P(H/S) = 2/3 = 66.7%

H S P(H IF S) P(H/S) = 2/3 = 66.7%

The probability that B happens IF A happens = the probability that A happens AND B happens the probability that A happens We rewrite this formula to obtain a formula for measuring the probability that A happens AND B happens

Multiply both sides of the equation by P(A):

There is a 40% chance that my car won’t start IF it rains. There is a 70% chance of rain. What is the probability that it rains AND my car won’t start. RC (.70) (.40) =.28

A jar contains 2 red and 3 blue marbles. Draw two without replacement. What is the probability they are both blue? P(both blue) = P(first is blue AND second is blue)

NOT OR IF AND