1. Learn to find the probability of independent and dependent events.

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. YOU TRY
Find the probability of rolling a six on the first roll of a 1-6 number cube and rolling an odd number of the second roll of the same cube.
The outcome of rolling a number cube does not
affect the outcome of rolling it again so the events
are independent.
P(6 and odd) = P(6) · P(odd)‏ 1 6 1 2 = 1 12
The probability of rolling a 6 and then rolling an odd is

9. 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

10. Example C
You are randomly choosing 2 cards from a regular deck of 52 cards. What is the probability you will pick an Ace and then a Jack without replacing the first card?
Pulling the first card changes the number of cards left, so the events are dependent.

11. P(Ace and then Jack) = P(A) · P(B after A)
Example C continued 4 52 1 13
There are 4 aces out of 52 cards.
P(Ace) = = 4 51
There are 4 jacks out of 51 cards.
(since one ace is already removed)
P(Jack) =
P(Ace and then Jack) = P(A) · P(B after A) = 1 13 4 51 4
663 =
Multiply.
The probability of choosing an ace and then a jack is 4
663

12. YOU Try
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.

13. 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

14. 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.

15. 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

16. Independent and Dependent Events
Pre-Algebra
10.6
Independent and Dependent Events
Extra Example
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 = = 16

17. Lesson Quiz: Part I
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

18. Lesson Quiz: Part II
3. Sarah picks 2 hats at random from 5 bill caps and 3 beanies. What is the probability that both are bill caps? 5 14