Probability and Odds

The probability of an event occurring is defined to be: # ways event can happen P(event) = # ways total( # way “anything” can happen) Remember that the probability of something occurring will always have a numerical value less than one Make sure you use the notation shown above – what do you want the probability OF?

The odds of an event occurring is the ratio of getting what you do want, to getting what you don’t want. # ways event WON’T happen Odds(event) = # ways event can happen Note the same notation

Roll a 6-sided die… 1 way to get a 3 6 ways total

Roll a 6-sided die… 1 way to get a 3 5 ways to get anything BUT a 3

Roll a 6-sided die… 4 ways to get less than a 5 (1, 2, 3, 4) 6 ways to get anything

Roll a 6-sided die… 4 ways to get less than a 5 2 ways to NOT get less than a 5 (5 or 6) Note that the value of odds CAN be greater than one, unlike probability. Also note that there are two standard ways of reporting odds – fractional (2/1) or using a colon as in ratios 2:1

If the odds against the Giants winning the pennant are 6 to 1, what is the probability that they will win it? 1 ways to win 6+1 ways to win OR lose P(win) =

Roll a 6-sided die… What if you roll the die 3 times in a row? P(three 4’s in a row)= Think of it as filling in the blanks for 3 events to occur.

The probability of consecutive events occurring (as above) is the product of each probability. We also call this “with replacement” because the 4 is “replaced” as a roll possibility each time the die is rolled. Another example is tossing a coin. The probability of consecutive events occurring (as above) is the product of each probability. We also call this “with replacement” because the 4 is “replaced” as a roll possibility each time the die is rolled. Another example is tossing a coin.

In the game of Blackjack, 2 cards are dealt from a standard, 52-card deck. The object is to get the closest to 21 points when the cards are added together.

What is the probability of getting two face cards (a great blackjack score) P(2 face cards)=P(1st card is face card) *P(2nd card face card) P(2 face cards)=P(1st card is face card) *P(2nd card face card) Think of it as filling in two slots Think of it as filling in two slots (one card has been removed) 12 total face cards 52 cards Note that the cards are NOT replaced when dealt, but order does not matter

4 cards are dealt in succession... What is the probability that the cards are all aces? P(all aces) =

4 cards are dealt in succession... What is the probability that the cards are NOT all aces? P(NOT all aces) = 1 - P(all aces)

4 cards are dealt in succession... What is the probability that the cards contain AT LEAST ONE ace? P(at least one ace)= P(1 ace)+P(2 aces)+P(3 aces)+P(4 aces) = 1-P(no aces)

P(at least one ace)=