Download Probability: Multiple Events (Dependent and Independent)

Name:
Date:
Algebra 2
Probability: Multiple Events (Dependent and Independent)
To find the probability that a 1st event happens and a 2nd event also happens, multiply the
two probabilities.
I. Dependent Events
When events are dependent, the second probability changes depending on what event happened first.
Tree diagrams for dependent events
To picture all the probabilities in situations involving successive events happening, it is helpful to
make a tree diagram showing all the possible sequences of events with their probabilities.
Example: This tree diagram represents choosing two students from a class of 13 girls (G) and 9 boys
(B). The top level of the tree shows the choice of the first student. The bottom level shows the choice
of a second student after the first student was chosen.
Note that all of the probabilities on the second level are out of 21, because there are 21 students
remaining after the first student has been chosen. Depending on the first choice made, the remaining
students are split either 12-and-9 or 13-and-8.
Name:
Date:
Algebra 2
1. Here is a tree diagram representing choosing two pieces of fruit from a refrigerator with 7 apples
and 5 pears.
a. Label every branch of this tree diagram with a probability.
b. Find the probability that the first piece of fruit is an apple and the second piece of fruit is a pear.
c. Find the probability that both pieces of fruit are pears.
2. There are 10 soda cans in a refrigerator: 3 regular sodas and 7 diet sodas. Suppose that two cans are
taken from the refrigerator for drinking, one after another.
a. Make a tree diagram representing this situation. Label every branch with its probability.
b. Using your tree diagram, answer the question: what is the probability that the two sodas taken
are both diet sodas?
c. Write a probability question about the sodas whose answer would be
3
10
 92 
6
90
.
Name:
Date:
Algebra 2
3. A deck of cards has 52 cards: 13 hearts (H) and 39 non-hearts (N). Two cards are dealt from the
deck, one card after the other.
a. Make a tree diagram representing this situation. Label every branch with its probability.
b. What is the probability that the first card is a heart but the second card is not a heart?
II. Independent Events
When events are independent, the second probability doesn’t change regardless of what happened in
the first event. In this case, finding the answer to an “and” probability problem is simpler: you just
find the two probabilities separately, then multiply.
Rule for independent events: If A and B are independent events, P(A and B) = P(A) · P(B)
4. One jar has 5 red marbles and 3 yellow marbles. Another jar has 4 green marbles and
6 blue marbles. Suppose that one marble is randomly drawn from each jar.
a. What is the probability of getting a red marble and a green marble?
b. What is the probability of getting a red marble and a blue marble?
c. Write a question about the marbles whose answer would be 83  106  18
80 .
Name:
Date:
Algebra 2
5. Suppose you roll two 6-sided dice. One die is red and the other die is green.
a. What is the probability of rolling a “3” on the red die and a “5” on the green die?
b. What is the probability of rolling a “3” on the green die and a “5” on the red die?
c. What is the probability of rolling a “3” and a “5” on the two dice?
Hint: Combine the answers from parts a and b.
d. What is the probability of rolling a “4” and a “4” on the two dice?
6. Probabilities of getting different color gumballs are given below. Answer these questions about
getting two gumballs from a machine. Assume the colors are independent (the color you get on the
first gumball does not affect the probabilities for the second gumball).
a. What is the probability that the first gumball is green and the second
gumball is blue?
b. What is the probability that both gumballs are white?
color
red
white
blue
pink
orange
green
gold
probability
0.2
0.15
0.1
0.27
0.07
0.2
0.01
c. What is the probability that the first gumball is red and the second gumball is not red?
d. What is the probability that both gumballs are gold? (If your calculator gives you an answer in
“E” notation, write it as an ordinary decimal.)
Name:
Date:
Algebra 2
Mixed Problems: dependent and independent
In the following, think carefully about whether the events involved are in each question are dependent
or independent.
7. Paige has 9 pens in her backpack: 6 blue and 3 red. Here are two slightly different questions about
the pens (the only difference is highlighted in bold).
a. Paige randomly takes a pen from her backpack to take notes in English.
She puts the pen away at the end of class.
Next period in Social Studies, again she randomly takes a pen from her backpack.
What is the probability that Paige used a red pen in English and a blue pen in Social Studies?
b. Paige randomly takes a pen from her backpack to take notes in English.
She forgets to put that pen away, leaving it on her English desk.
Next period in Social Studies, again she randomly takes a pen from her backpack.
What is the probability that Paige used a red pen in English and a blue pen in Social Studies?
8. A child’s toy box contains 7 rectangle blocks and 5 triangle blocks.
a. Suppose that a block is taken from the toy box, and not put back.
Then, another block is taken from the toy box.
What is the probability that a triangle block was taken the first time and a rectangle block was
taken the second time?
b. Suppose that a block is taken from the toy box, and then returned.
Then, a block is taken from the toy box again.
What is the probability that a triangle block was taken the first time and a rectangle block was
taken the second time?
Name:
Date:
Algebra 2
9. Here is some given information about events A, B, C, and D:
□
□
□
□
□
□
P(A) = 0.4
P(A or B) = 0.7
P(A and B) = 0
P(C) = 0.2
If event C occurs, then the probability of D is 0.65
If event C does not occur, then the probability of D is 0.8
a. Are events A and B mutually exclusive? Explain why or why not.
b. Are events C and D independent? Explain why or why not.
c. Find P(B).
d. Find the probability that neither A nor B occurs.
e. Find P(C and D).
Name:
Date:
Algebra 2
Answers: “Probability: Dependent and Independent”
1.
7
5
12 12
, ; 116 , 115 , 117 , 114 b.
7
12
35
 115  132
20
c. 125  114  132
2. a. Tree showing these probabilities:
3 7
10 10
, ; 92 , 79 , 93 , 96
7
10
b.
 96 
42
90
c. When choosing one soda followed by another, what is the prob. that both sodas are regular?
3. a. Tree showing these probabilities:
c.
39
52
 38
51 
4. a.
5
8
 104 
5. a.
1
6
 16 
6. a. 0.02
13
52
12 39 13 38
, 39
52 ; 51 , 51 , 51 , 51
13
52
b.
 39
51 
507
2652
1482
2652
20
80
1
36
b.
5
8
 106 
b.
1
6
 16 
b. 0.0225
1
36
c. 0.16
 96  18
81
7. a. independent…
3
9
8. a. dependent…
35
 117  132
5
12
30
80
c. What is the prob. of getting a yellow and a blue?
c.
1
36
 361  181
1
6
d.
 16 
1
36
d. 0.01  0.01 = .0001
b. dependent…
3
9
b. independent…
 86 
5
12
18
72
35
 127  144
9. a. Yes, because P(A and B) = 0.
b. No, because the probability of event D is different depending on whether C occurs or not.
c. Can use P(A or B) = P(A) + P(B) because the events are mutually exclusive, so P(B) = 0.3.
d. 1 – P(A or B) = 1 – 0.7 = 0.3.
e. P(C and D) = P(C) · P(D given that C happens) = 0.2 · 0.65 = 0.13.