A poker hand consists of five cards. A. Find the total number of possible five-card poker

hands.

B. Find the number of ways in which four aces can be selected.

C. Find the number of ways in which one king can be selected.

D. Use the FCP and previous answers to find the number of ways of getting four aces and one king.

E. Find the probability of getting a poker hand of four aces and one king.

Thinking Mathematically

I can find the probability that an event will not occur.

I can find the probability of one event or a second event occurring.

I can use odds.

• A survey asked 500 Americans to rate their health. Of those surveyed, 270 rated their health as good/excellent. This means that 500 – 270, or 230, people surveyed did not rate their health as good/excellent.

P(good/excellent) + P(not good/excellent) =

P(E) + P(not E) = 1

270 230 5001

500 500 500

The Probability of an Event Not Occurring

The probability that an event E will not occur is equal to 1 minus the probability that it will occur.

P(not E) = 1 - P(E)

• If you are dealt one card from a standard 52-card deck, find the probability that you are NOT dealt a queen.

P(not E) = 1 – P(E) so

P(not a queen) = 1 – P(queen).

There are four queens in a deck of 52 cards. The probability of being dealt a queen is 4/52 or 1/13.

So, P(not a queen) = 1 – 1/13 or

12/13

You try• If you are dealt one card from a standard

52-card deck, find the probability that you are not dealt a diamond.

• If you are dealt one card from a standard 52-card deck, find the probability that you are not dealt a black two.

Mutually Exclusive Events

If it is impossible for events A and B to occur simultaneously, the events are said to be mutually exclusive.

If A and B are mutually exclusive events, thenP(A or B) = P(A) + P(B).

Or Probabilities with Events That Are Not Mutually Exclusive

If A and B are not mutually exclusive events, then

P(A or B) = P(A) + P(B) - P(A and B)

• In a group of 25 baboons, 18 enjoy grooming their neighbors, 16 enjoy screeching wildly, while 10 enjoy grooming their neighbors and screeching wildly. If one baboon is selected at random from the group, find the probability that it enjoys grooming its neighbors or screeching wildly.

• It is possible for a baboon in the group to enjoy both grooming its neighbors and screeching wildly. Ten of the brutes are given to engage in both activities. These events are NOT mutually exclusive.

P(grooming or screeching) = P(grooming) + P(screeching) – P(grooming or screeching)

18/25 + 16/25 – 10/25 = (18 + 16 – 10)/25 =

24/25

You try• In a group of 50 students, 23 take math, 11 take

psychology, and 7 take both math and psychology. If one student is selected at random, find the probability that the student takes math or psychology.

• If one person is randomly selected from the U.S. military, find the probability, using the table, that this person is in the Army or is a woman.

Air Force

Army Marine Corps

Navy Total

Male 290 400 160 320 1170

Female 70 70 10 50 200

Total 360 470 170 370 1370

• It is possible to select a person who is both in the Army and is a woman. So, these events are not mutually exclusive.

P(Army or woman) =

P(Army)+P(woman)-P(Army and woman)

= 470/1370 + 200/1370 – 70/1370

= (470 + 200 – 70)/1370

= 600/1370

= 60/137

Probability to OddsIf P(E) is the probability of an event E occurring, then1. The odds in favor of E are found by taking the

probability that E will occur and dividing it by the probability that E will not occur.Odds in favor of E = P(E) / P(not E)

2. The odds against E are found by taking the probability that E will not occur and dividing by the probability that E will occur.Odds against E = P(not E) / P(E)

The odds against E can also be found by reversing the ratio representing the odds in favor of E.

You roll a single, six-sided die. • Find the odds in favor of rolling a 2.

• Let E represent the event of rolling a 2. The total number of possibilities is 6. So, P(E) = 1/6 and the

P(not E) = 1 – 1/6 or 5/6 So, the odds in favor of E

(rolling a 2) = P(E)/P(not E) = = =

1656

1 6

6 5

1

5

You roll a single, six-sided die. • Find the odds against rolling a 2.

• Now that we know the odds in favor of rolling a 2, 1:5 or 1/5, we can find the odds against rolling a 2 by reversing this ratio. Thus,

Odds against E (rolling a 2) = or 5

1 5 :1

You try one• You are dealt one card from a 52-card deck.

Find the odds in favor of getting a red queen.

Find the odds against

getting a red queen.

• The winner of a raffle will receive a new sports utility vehicle. If 500 raffle tickets were sold and you purchased ten tickets, what are the odds against your winning the car?

• Let E represent the event of winning the SUV. Because you purchased ten tickets and 500 were sold.

P(E) = 10/500 = 1/50 and P(not E) = 1 – 1/50 = 49/50

Odds against E = P(not E)/P(E) = = =

49501

50

49

149 :1

The winner of a raffle ticket will receive a two-year scholarship to the college of his or her choice. If 1,000

raffle tickets were sold and you purchased five tickets, what are the

odds against your winning the scholarship?

Odds to Probability

If the odds in favor of an event E are a to b, then the probability of the event is given by

P(E) = a

a+b

The odds in favor of a particular horse winning a race are 2 to 5. What is the probability that this horse will win the

race? • Because odds in favor, a to b, means a probability of

a/(a+b), then odds in favor, 2 to 5, means a probability of =

The probability that this horse will win the race is

2

2 52

7

2

7

• The odds against a particular horse winning a race are 15 to 1. Find the odds in favor of the horse winning the race and the probability of the horse winning the race.

Thinking Mathematically

Events Involving Not and Or; Odds

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