1. To find the probability of independent events dependent events
Statistics and Probability
To find the probability of
independent events
dependent events
Course 2, Lesson 9-7

2. Raji and Kara must each choose a solo from a list of music pieces to play for their recital.
If Raji’s choice has no effect on Kara’s choice and vice versa, the events are independent. For independent events, the occurrence of one event has no effect on the probability that a second event will occur.

3. If once Raji chooses a solo, Kara must choose from the remaining solos, then the events are dependent. For dependent events, the occurrence of one event does have an effect on the probability that a second event will occur.

4. Example A
Decide whether the set of events are dependent or independent. Explain your answer.
1. Erika rolls a 3 on one number cube and a 2 on another number cube.
Independent – rolling the first cube does not affect the outcome of rolling the second cube
2. Tomoko chooses a seventh-grader for her term from a group of seventh- and eighth-graders, and then Juan chooses a different seventh-grader from the remaining students.
Dependent – Tomoko’s first choice affects the outcome of Juan’s choice

5. P(A and B)‏ P(A)‏ P(B)‏ = Probability of Two Independent Events
To find the probability that two independent events will happen, multiply the probabilities of the two events.
Probability of Two Independent Events
P(A and B)‏ =
P(A)‏
P(B)‏ •
Probability of
both events
Probability of
first event
Probability of
second event

6. Example B
Find the probability of flipping a coin and getting heads and then rolling a 6 on a number cube.
The outcome of flipping a coin does not affect the outcome of rolling a number cube so the events are independent.
P(heads and 6) = P(heads) · P(6)‏ 1 2 = 6 1 12 = 1 12
The probability of flipping a heads and rolling a 6 is

7. YOU TRY
Find the probability of choosing a red marble at random from a bag containing 5 red and 5 white marbles and then flipping a coin and getting heads.
The outcome of choosing the marble does not affect
the outcome of flipping the coin, so
the events are independent.
P(red and heads) = P(red) · P(heads)‏ 1 2 1 2 =
The probability of choosing a red marble and a
coin landing on heads is 1 4

8. P(A and B)‏ P(A)‏ P(B after A)‏ = Probability of Two Dependent Events
To find the probability of two dependent events, you must determine the effect that the first event has on the probability of the second event.
Probability of Two Dependent Events
P(A and B)‏ =
P(A)‏
P(B after A)‏ •
Probability of
both events
Probability of
first event
Probability of
second event

9. Example C
A reading list contains 5 historical books and 3 science-fiction books. What is the probability that Juan will randomly choose a historical book for his first report and a science-fiction book for his second?
The first choice changes the number of books left, and may change the number of science-fiction books left, so the events are dependent.

10. P(historical and then science-fiction) = P(A) · P(B after A)
You try Continued 5 8
There are 5 historical books out of 8 books.
P(historical) = 3 7
There are 3 science-fiction books left
out of 7 books.
P(science-fiction) =
P(historical and then science-fiction) = P(A) · P(B after A) = 5 8 3 7 15 56 =
Multiply.
The probability of Juan choosing a historical book and
then choosing a science-fiction book is 15 56

11. You try
Alice was dealt a hand of cards consisting of 4 black and 3 red cards. Without seeing the cards, what is the probability that the first card will be black and the second card will be red?
The first choice changes the total number of cards left, and may change the number of red cards left, so the events are dependent.

12. You try Continued 4 7
P(black) =
There are 4 black cards out of 7 cards. 3 6
There are 3 red cards left out of 6 cards.
P(red) =
P(black and then red card) = P(A) · P(B after A) = 4 7 3 6 12 42 2 7 = or
Multiply.
The probability of Alice selecting a black card and
then choosing a red card is . 2 7

13. Independent and Dependent Events
Pre-Algebra
10.6
Independent and Dependent Events
Example e
The letters in the word dependent are placed in a box.
A. If two letters are chosen at random and not replaced, what is the probability that they will both be consonants? 69 = 23
P(first consonant) = 58
P(second consonant) = 23 58
10 24
5 12 = = 13

14. The two spinners below are spun. What is the
Need Another Example?
The two spinners below are spun. What is the
probability that both spinners will show a
number greater than 6?
Answer

15. P(selecting a vowel) = or . P(spinning a vowel) = .
Step-by-Step Example 1.
One letter tile is selected and the spinner is spun. What is the probability that both will be a vowel?
There are 12 outcomes. Two outcomes contain only vowels. The probability that both will be a vowel is or . 1
Make a Tree Diagram 2
Tile
Spinner
Sample Space G B E A A B C
G, A
G, B
G, C
B, A
B, B
B, C
E, A
E, B
E, C
A, A
A, B
A, C 3
Use Multiplication
P(selecting a vowel) = or .
P(spinning a vowel) = .
P(both vowels) = • or .
Multiply the probabilities. 4
So, using either method the probability is .
Need Another Example?

16. First, find the probability of each event.
Step-by-Step Example 2.
The spinner and number cube shown are used in a game. What is the probability of a player not spinning blue and then rolling a 3 or 4?
You are asked to find the probability of the spinner not landing on blue and rolling a 3 or 4 on a number cube. The events are independent because spinning the spinner does not affect the outcome of rolling a number cube. 1
First, find the probability of each event. 2
P(not blue) =
P(3 or 4) = or 3
Then, find the probability of both events occurring.
P(not blue and 3 or 4) =
P(A and B) = P(A) · P(B) =
Multiply. 4
The probability is .
Check Make an organized list, table, or a tree
diagram to show the sample space. 5
Need Another Example?

17. Decide whether each event is independent or dependent. Explain.
1. Mary chooses a game piece from a board game, and then Jason chooses a game piece from three remaining pieces.
2. Find the probability of spinning an evenly divided spinner numbered 1–8 and getting a composite number on one spin and getting an odd number on a second spin.
Dependent; Jason has
fewer pieces from which to choose. 3 16 5 14