Transcript odds against E

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.
270 230 500


1
P(good/excellent) + P(not good/excellent) =
500 500 500
P(E) + P(not E) = 1
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, then
P(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 Navy
Corps
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 Odds
If P(E) is the probability of an event E occurring,
then
1. 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
1
1 6
6
(rolling a 2) = P(E)/P(not E) = 5 =  =
6 5
6
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) =
5
1
or
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
49
49
50
Odds against E = P(not E)/P(E) =
=
=
1
1
50
49 :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 2 = 2
25
7
The probability that this horse will win the race is
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
Have a wonderful Day!!!