Download 9.7 - Probability of Multiple Events

9.7 – Probability of
Multiple Events
Warm Up
(For help, go to Lesson 1-6.)
A bag contains 24 green marbles, 22 blue marbles, 14 yellow marbles,
and 12 red marbles. Suppose you pick one marble at random.
Find each probability.
1. P(yellow)
2. P(not blue)
3. P(green or red)
Warm Up - Solutions
1. total number of marbles = 24 + 22 + 14 + 12 = 72
P(yellow) =
2•7
7
14
= /
=
36
72 2/ • 36
2. total number of marbles = 24 + 22 + 14 + 12 = 72
P(not blue) =
72 – 22 50 2/ • 25
25
=
=
=
72 2/ • 36
72
36
3. total number of marbles = 24 + 22 + 14 + 12 = 72
P(green or red) =
1 • 36
24 + 12
= 36 =
=1
2 • 36
72
2
72
Warm Up
Of 300 senior students at Howe High,
150 have taken physics,
192 have taken chemistry, and 30
have taken neither physics nor
chemistry. How many students have
taken both physics and chemistry?
Warm Up - Solutions
Let x = the number of students who have taken both physics and chemistry.
Then (150 – x) is the number of students who have taken physics,
but not chemistry. And (192 – x) is the number of students who have
taken chemistry, but not physics. 30 students have taken neither, and
there are 300 students altogether.
x + (150 – x) + (192 – x) + 30 = 300
(150 + 192 + 30) + (1 – 1 – 1)x = 300
372 – x = 300
372 – 300 = x
72 = x
So, 72 students have taken both physics and chemistry.
Consider the Following:
A
marble is picked at random from a
bag. Without putting the marble
back, a second one has chosen. How
does this affect the probability?
 A card is picked at random from a
deck of cards. Then a dice is rolled.
How does this affect the probability?
Outcomes of Different Events
 When
the outcome of one event
affects the outcome of a second
event, we say that the events are
dependent.
 When one outcome of one event
does not affect a second event, we
say that the events are independent.
Probability of Multiple Events
Classify each pair of events as dependent or independent.
a. Spin a spinner. Select a marble from a bag that contains marbles
of different colors.
Since the two events do not affect each other, they are independent.
b. Select a marble from a bag that contains marbles of two colors.
Put the marble aside, and select a second marble from the bag.
Picking the first marble affects the possible outcome of picking the
second marble. So the events are dependent.
Decide if the following are
dependent or independent
 An
expo marker is picked at
random from a box and then
replaced. A second marker is
then grabbed at random.
 Two dice are rolled at the
same time.
 An Ace is picked from a deck
of cards. Without replacing it,
a Jack is picked from the
deck.
Independent
Independent
Dependent
How to find the Probability of Two
Independent Events
 If
A and B are independent events,
the P(A and B) = P(A) * P(B)
 Ex:
If P(A) = ½ and P(B) = 1/3 then
P(A and B) =
1 1 1
 
2 3 6
Let’s Try One
A box contains 20 red marbles and 30 blue marbles. A second box contains 10
white marbles and 47 black marbles. If you choose one marble from each box
without looking, what is the probability that you get a blue marble and a black
marble?
Relate: probability of both events is probability of first event
times probability of second event
30
Define: Event A = first marble is blue. Then P(A) = 50.
Event B = second marble is black. Then P(B) = 47.
57
Write: P(A and B) = P(A) • P(B)
30
47
P(A and B) = 50 • 57
= 1410 .
2850
=
47
95
Simplify.
47
The probability that a blue and a black marble will be drawn is 95 , or 49%.
Mutually Exclusive Events
 Two
events are mutually exclusive
then they can not happen at the
same time.
Probability of Multiple Events
Are the events mutually exclusive? Explain.
a. rolling an even number or a prime number on a number cube
By rolling a 2, you can roll an even number and a prime number
at the same time.
So the events are not mutually exclusive.
b. rolling a prime number or a multiple of 6 on a number cube
Since 6 is the only multiple of 6 you can roll at a time and it is not a prime
number, the events are mutually exclusive.
How to find the Probability of Two
Mutually Exclusive Events
 If
A and B are mutually exclusive
events, then P(A or B) = P(A) + P(B)
 If
A and B are not mutually exclusive
events, then
P(A or B) = P(A) + P(B) – P(A B)
Let’s Try Some
A
spinner has ten equal-sized
sections labeled 1 to 10. Find the
probability of each event.
A)P(even or multiple 0f 5)
B)P(Multiple of 3 or 4)
Hint: Decide if each
event is mutually
exclusive
No!
P(A)+P(B)-P(A B)
Yes!
P(A)+P(B)
Let’s Try Some
A)P(even or multiple 0f 5)
B)P(Multiple of 3 or 4)
Let’s Try One
At a restaurant, customers get to choose one of
four desserts. About 33% of the customers
choose Crème Brule, and about 28% Chocolate
Cheese Cake. Kayla is treating herself for pole
vaulting four feet at the meet. What is the
probability that Kayla will choose Crème Brule
or Chocolate Cheese Cake?
Are the events mutually exclusive?
Solution:
.33 + .28 = .61 = 61%
Yes. So:
P(A) + P(B)
Probability of Multiple Events
A spinner has twenty equal-size sections numbered from 1 to 20. If
you spin the spinner, what is the probability that the number you spin
will be a multiple of 2 or a multiple of 3?
No. So:
Are the events mutually exclusive?
P(A) + P(B) P(AB)
P(multiple of 2 or 3) = P (multiple of 2) +
P (multiple of 3) – P (multiple of 2 and 3)
10
6
3
= 20 + 20 – 20
=
13
20
13
The probability of spinning a multiple of 2 or 3 is 20.