11.2
Probability of Compound Events
Lesson Objective
• Use possibility diagrams to find probability of compound events.
Use Possibility Diagrams to Find Probability of Compound Events.
You have seen how to find all possible outcomes of a compound event by drawing a
possibility diagram. From a possibility diagram, you are able to count the number of
favorable outcomes and the total number of outcomes.
You can then find the probability of a compound event E using the definition of
probability:
P(E ) 5
Number of outcomes favorable to event E
Total number of equally likely outcomes
You have learned how to draw a tree diagram to represent the possible outcomes
of tossing coins. Consider finding the probability of landing heads, then tails when
tossing a coin twice.
1st Toss
2nd Toss
Outcome
H
(H, H)
T
(H, T)
H
(T, H)
H
T
T
(T, T)
H represents heads
T represents tails
You can see that there are four possible outcomes: (H, H), (H, T), (T, H), and (T, T).
However, there is only one favorable outcome, (H, T).
So, P(H, T) 5
1
4
You want to find the probability
of landing heads then tails, so the
order of events is important. P(H, T)
is not the same as P(T, H) here.
Lesson 11.2 Probability of Compound Events
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Example 4
Use a possibility diagram to find the probability of a
compound event.
Use a possibility diagram to find each probability.
a)
A fair coin and a fair six-sided number die are tossed together. Find the
probability of showing heads and a 5.
Solution
Method 1
T
Coin
Tossed
H
1
2
3
4
5
H represents heads
T represents tails
6
Number Die Rolled
P(H, 5) 5
1
12
Think Math
Would the probability be different if
the possibility diagram were drawn
with the axis labels interchanged?
Explain your answer.
You can see that there
is only one outcome
with heads and a 5.
Method 2
Coin
Number Die
1
2
3
4
5
6
H
(1, H)
(2, H)
(3, H)
(4, H)
(5, H)
(6, H)
T
(1, T)
(2, T)
(3, T)
(4, T )
(5, T )
(6, T )
H represents heads
T represents tails
P(H, 5) 5 P(5, H)
1
5
12
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b)
Two fair six-sided number dice are rolled. Find the probability that the sum of
the two numbers rolled is a prime number.
Solution
2nd Number Die
1st Number Die
Math Note
+
1
2
3
4
5
6
1
2
3
4
5
6
7
2
3
4
5
6
7
8
3
4
5
6
7
8
9
4
5
6
7
8
9
10
5
6
7
8
9
10
11
6
7
8
9
10
11
12
Recall that a prime number is a whole
number greater than 1 that is divisible
by only 1 and itself.
The prime numbers in the possibility
diagram are 2, 3, 5, 7, and 11. There
are 15 prime numbers out of 36
equally likely possible outcomes.
P(sum is prime) 5
15
36
5
5
12
c)
Two fair four-sided number dice, one red (R) and one blue (B), are rolled, and the
number on the bottom is recorded. The red number die has numbers 1, 2, 4, and
7. The blue number die has numbers 2, 5, 8, and 9. Find the probability that the
number recorded from the blue number die is more than 3 greater than the
number recorded from the red number die. That is, find P(B 2 R  3).
Solution
Method 1
Blue Number Die
Red Number Die
1
2
4
7
2
(1, 2)
(2, 2)
(4, 2)
(7, 2)
5
(1, 5)
(2, 5)
(4, 5)
(7, 5)
8
(1, 8)
(2, 8)
(4, 8)
(7, 8)
9
(1, 9)
(2, 9)
(4, 9)
(7, 9)
P(B 2 R  3) 5
7
16
Continue on next page
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Method 2 You can list the differences
instead of the numbers
Blue Number Die
Red Number Die
rolled. Then look for
2
1
2
4
7
2
1
0
22
25
5
4
3
1
22
8
7
6
4
1
9
8
7
5
2
P(B  R  3) 5
differences greater than 3.
7
16
d) The two spinners shown are spun. Find the
probability that the pointers stop at 1 on Spinner 1
and blue (B) on Spinner 2.
1
1
2
4
B
(1, B)
(1, B)
(2, B)
(4, B)
G
(1, G)
(1, G)
(2, G)
(4, G )
R
(1, R)
(1, R)
(2, R)
(4, R )
G
2
Spinner 1
Spinner 1
Spinner 2
1
B
4
Solution
P(1, B) 5
1
R
Spinner 2
2
1
5
12
6
Guided Practice
Draw a possibility diagram to find each probability.
1 Two fair four-sided number dice, each numbered 1 to 4, are rolled together. The
result recorded is the number facing down. Find the probability that the product
of the two numbers is divisible by 2.
2 One colored disc is randomly drawn from each of two bags. Each bag has
5 colored discs: 1 red, 1 green, 1 blue, 1 yellow, and 1 white. Find the probability
of drawing a blue or yellow disc.
3 A box has 1 black, 1 green, 1 red, and 1 yellow marble. Another box has 1 white,
232
1 green, and 1 red marble. A marble is taken at random from each box. Find the
probability that a red marble is not drawn.
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Example 5
Use tree diagrams to find probabilities of compound events.
Suppose that it is equally likely to rain or not rain on any given day. Draw a
tree diagram and use it to find the probability that it rains exactly once on two
consecutive days.
1st Day
2nd Day
Outcome
R
(R, R)
R9
(R, R9)
R
(R9, R)
R
R9
R9
(R9, R9)
R represents raining
R9 represents not raining
The favorable outcome is “rain exactly once
on two consecutive days.” So, you should look
for the outcome that gives (R, R9) or (R9, R),
meaning either it rains on the first day or it
rains on the second day.
Solution
P(rain exactly once on two consecutive days) 5
2
4
5
1
2
1
2
So, the probability of (R, R′) or (R′, R) is .
Guided Practice
Solve. Show your work.
4 Three fair coins are tossed together.
a) Draw a tree diagram to represent all possible outcomes.
b) Using your answer in a), find the probability of getting all heads.
c)
Using your answer in a), find the probability of getting at least two tails.
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Practice 11.2
Solve. Show your work.
1 A bag contains 2 blue balls and 1 red ball. Winnie randomly draws a ball from the
bag and replaces it before she draws a second ball. Use a possibility diagram to
find the probability that the balls drawn are different colors.
2 A letter is randomly chosen from the word FOOD, followed by randomly choosing
a letter from the word DOG. Draw a tree diagram to find the probability that both
letters chosen are the same.
3 Three pebbles are placed in a bag: 1 blue, 1 green, and 1 yellow. First a pebble
is randomly drawn from the bag. Then a fair four-sided number die labeled
1 to 4 is rolled. The result recorded is the number facing down. Use a possibility
diagram to find the probability of drawing a yellow pebble and getting a 4.
4 Thomas rolled two number dice: a fair six-sided one and a fair four-sided one labeled
1 to 4. Use a possibility diagram to find the probability of rolling the number 3
on both dice.
5 A bike shop has 3 bikes with 20-speed gears and 2 bikes with 18-speed gears. The
bike shop also sells 1 blue helmet and 1 yellow helmet. Use a possibility diagram to
find the probability of getting an 18-speed bicycle and a blue helmet if one bike and
one helmet is randomly selected from the shop.
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6 Susan randomly draws a card from three number cards: 1, 3, and 6. After replacing
the card, Susan randomly draws another number card. The product of the two
numbers drawn is recorded.
1
3
6
a) Draw a possibility diagram to represent all possible outcomes.
b) Using your answer in a), what is the probability of forming a number greater
than 10 but less than 30?
7 Jane and Jill watch television together for 2 hours. Jane selects the channel for the
first hour, and Jill selects the channel for the second hour. Jane’s remote control
randomly selects from Channels A, B, and C. Jill’s remote control randomly selects
from Channels C, D, and E.
a) Draw a possibility diagram to represent all possible outcomes for the channels
they watch on television for 2 hours.
b) Using your answer in a), what is the probability of watching the same channel
for both hours?
8 A color disc is randomly drawn from a bag that contains the following discs.
Y
R
B
R
G
After a disc is drawn, a fair coin is tossed. Use a possibility diagram to find the
probability of drawing a red disc and landing on heads.
9 Karen tosses a fair coin three times. Draw a tree diagram to find the probability of
getting the same result in all three tosses.
10 Mrs. Bridget wants to make pancakes. She has 3 types of flour: cornmeal flour,
whole wheat flour, and white flour. She also has 2 cartons of milk: whole milk and
low fat milk. However, all the types of flour and milk are not labeled, so she
randomly guesses which types of flour and milk to use for her pancakes.
a) Draw a tree diagram to represent all possible outcomes.
b) Using your answers in a), find the probability that she will use cornmeal flour and
low fat milk for her pancakes.
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