Probability and Odds Probability and Odds

The probability of an event occurring is defined to be:

# ways event can happenP(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…Roll a 6-sided die…

1 way to get a 3

6 ways total 6

1)3( arollP

Roll a 6-sided die…Roll a 6-sided die…

1 way to get a 3

5 ways to get anything BUT a 3

5

1)3( arollodds

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

6 ways to get anything

3

2

6

4)5( athanlessP

Roll a 6-sided die…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)

1:21

2

2

4)5( thanlessodds

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 If the odds against the Giants winning the pennant are 6 to 1, winning the pennant are 6 to 1, what is the probability that what is the probability that they will win it?they will win it?

1 ways to win

6+1 ways to win OR lose

P(win) =

Roll a 6-sided die…Roll a 6-sided die…

216

1

6

1

)4()4()4(

3

arollParollParollP

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

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

What is the probability of What is the probability of getting two face cards (a great getting two face cards (a great blackjack score)blackjack score)

• P(2 face cards)=P(1st card is face P(2 face cards)=P(1st card is face card) *P(2nd card face card) card) *P(2nd card face card)

• Think of it as filling in two slotsThink of it as filling in two slots

221

11

51

11*

52

12

(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. . .4 cards are dealt in succession. . .

What is the probability that the cards What is the probability that the cards are all aces?are all aces?

P(all aces) =

452

44

25

1

17

1

13

1

725,270

1

49

1*

50

2*

51

3*

52

4

C

C

4 cards are dealt in succession. . . 4 cards are dealt in succession. . .

What is the probability that the cards What is the probability that the cards are NOT all aces?are NOT all aces?

P(NOT all aces) = 1 - P(all aces)

725,270

724,270

725,270

11

4 cards are dealt in succession. . . 4 cards are dealt in succession. . .

What is the probability that the cards What is the probability that the cards contain AT LEAST ONE ace?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)

49*50*51*52

45*46*47*481

!52!44

!48!481

!4!48!52!4!44

!48

11452

448 C

C

49*5*17*13

9*46*47*21

49*50*51*52

45*46*47*481

281.054145

15229

P(at least one ace)=