Independent and Independent and Dependent EventsDependent Events

Goal: Goal: To find the probability of two To find the probability of two independentindependent or two or two dependent dependent events.events.

Independent EventsIndependent Events

Events for which the occurrence of the Events for which the occurrence of the FIRST DOES NOT effect that of the SECOND FIRST DOES NOT effect that of the SECOND because:because:

1.1.The two events are The two events are unrelatedunrelatedOROR

2.2.You repeat an event with an item whose You repeat an event with an item whose numbers will not change (ex: spinners or numbers will not change (ex: spinners or dice)dice)

OROR3.3.You repeat the same activity, but you You repeat the same activity, but you

REPLACEREPLACE the item that was removed. the item that was removed.

The two events are unrelatedThe two events are unrelated

P(choosing a jack, rolling a 2)P(choosing a jack, rolling a 2) P (spinning a 3, pulling an ace)P (spinning a 3, pulling an ace) P (rolling a 6, spinning a red)P (rolling a 6, spinning a red)

Independent Events #1Independent Events #1

Independent Events #2Independent Events #2

You repeat an event with an item whose You repeat an event with an item whose numbers will not change (e.g.: numbers will not change (e.g.: spinners, dice, flipping a coin, etc.)spinners, dice, flipping a coin, etc.)

Independent Events #3Independent Events #3

You repeat the same activity by You repeat the same activity by pulling something out, but you pulling something out, but you REPLACEREPLACE the item that was the item that was removed.removed.

Probability of Independent Probability of Independent EventsEvents

When you have two independent events:When you have two independent events: Find the probability of the Find the probability of the firstfirst event event Find the probability of the Find the probability of the secondsecond event event MultiplyMultiply the probabilities together the probabilities together ReduceReduce your answer your answer

For independent events A and B, the probability of For independent events A and B, the probability of both events occurring is found by multiplying the both events occurring is found by multiplying the probabilities of the events.probabilities of the events.

P(A and B) = P(A and B) = P(A) P(A) • P(B)• P(B)

An Independent EventAn Independent EventThere are 10 socks in a basket: 5 blue, 3 yellow and 2 There are 10 socks in a basket: 5 blue, 3 yellow and 2 red. What is the probability of choosing a yellow sock red. What is the probability of choosing a yellow sock and a red sock if the sock and a red sock if the sock is replaced after the first is replaced after the first event?event?

is replaced

P(yellow, red)

Yellow:

Replace the yellow sock.

Red:

33

1010

22

1010=

11

55

x

33

5050

Probability of Independent Probability of Independent EventsEvents

A bag containing 4 green marbles, 6 red marbles, and 2 white A bag containing 4 green marbles, 6 red marbles, and 2 white marbles. Three marbles are drawn at random with marbles. Three marbles are drawn at random with replacement. replacement. With replacement With replacement means that after a marble means that after a marble is drawn, it is replaced before the next one is drawn. Find is drawn, it is replaced before the next one is drawn. Find each probability.each probability.

1.1. P(green, red, white)P(green, red, white)

2.2. P(green, white, white)P(green, white, white)

3.3. P(all three red)P(all three red)

4.4. P(not green, not red, not white)P(not green, not red, not white)

Dependent EventsDependent Events

Events for which the occurrence of the Events for which the occurrence of the FIRST AFFECTS that of the SECONDFIRST AFFECTS that of the SECOND

Look for the following 2 clues to Look for the following 2 clues to determine if an event is dependentdetermine if an event is dependent

You remove something after the first event and You remove something after the first event and DO NOT REPLACEDO NOT REPLACE it. it.

You ask yourself the question, You ask yourself the question, ““What happens What happens second?second?”” and your answer is and your answer is ““It It dependdepends...s...””

Probability of Dependent EventsProbability of Dependent Events When you have two dependent events:When you have two dependent events:

Find the probability of the Find the probability of the firstfirst event event Figure out Figure out whatwhat has has changedchanged

Numerator and/or denominatorNumerator and/or denominator Figure out the probability of the second eventFigure out the probability of the second event MultiplyMultiply the 2 probabilities together the 2 probabilities together ReduceReduce your answer your answer

For dependent events A and B, the probability of For dependent events A and B, the probability of both events occurring is found by multiplying the both events occurring is found by multiplying the probabilities of the events.probabilities of the events.

P(A and B) = P(A and B) = P(A) P(A) • P(B)• P(B)

A Dependent EventA Dependent EventThere are 10 socks in a basket: 5 blue, 3 yellow and 2 There are 10 socks in a basket: 5 blue, 3 yellow and 2 red. What is the probability of choosing a yellow sock red. What is the probability of choosing a yellow sock and a red sock if the sock and a red sock if the sock is NOT replaced after the is NOT replaced after the first event?first event?

is NOT replaced

P(yellow, red)

Yellow:

DO NOT replace the yellow sock.

Red:

33

1010

22

99

x

66

9090

11

1515=

Notice that the denominator

changed

Probability of Dependent EventsProbability of Dependent EventsAt a meeting there are 6 juniors and 12 seniors. Four At a meeting there are 6 juniors and 12 seniors. Four people are selected at random, one at a time for a people are selected at random, one at a time for a committee. Find each probability.committee. Find each probability.

1.1. P(all juniors)P(all juniors)

2.2. P(junior, junior, senior, senior)P(junior, junior, senior, senior)

3.3. P(senior, senior, senior, junior)P(senior, senior, senior, junior)

4.4. P(all senior)P(all senior)

TEST YOURSELFTEST YOURSELF

Are these dependent or Are these dependent or independent events?independent events?

Are these dependent or Are these dependent or independent events?independent events?

Tossing two dice and getting a 6 on Tossing two dice and getting a 6 on both of them. both of them.

Independent Event

Are these dependent or Are these dependent or independent events?independent events?

You pick the letter Q from a bag containing You pick the letter Q from a bag containing all the letters of the alphabet. You do not all the letters of the alphabet. You do not put the Q back in the bag before you pick put the Q back in the bag before you pick another tile. another tile.

Dependent Event

Are these dependent or Are these dependent or independent events?independent events?

You have a bag of marbles: 3 blue, You have a bag of marbles: 3 blue, 5 white, and 12 red. You choose 5 white, and 12 red. You choose one marble out of the bag, look at it one marble out of the bag, look at it then put it back. Then you choose then put it back. Then you choose another marble. another marble.

Independent Event

Are these dependent or Are these dependent or independent events?independent events?

You pick the letter W from a bag You pick the letter W from a bag containing all the letters of the alphabet. containing all the letters of the alphabet. You put the W back in the bag and pick a You put the W back in the bag and pick a second time. second time.

Independent Event

Are these dependent or Are these dependent or independent events?independent events?

You have a basket of socks. You You have a basket of socks. You need to find the probability of need to find the probability of pulling out a black sock and its pulling out a black sock and its matching black sock without matching black sock without putting the first sock back.putting the first sock back.

Dependent Event

Find the probabilityFind the probability

P(jack, green)P(jack, green)11

55

22

88x =

22

4040

11

2020

Find the probabilityFind the probability

P(6, not 5)P(6, not 5)11

66

55

66x =

55

3636

Find the probabilityFind the probability

P(Q, Q)P(Q, Q) All the letters of All the letters of

the alphabet are in the alphabet are in the bag 1 timethe bag 1 time

Do not replace the Do not replace the letterletter

11

2626

00

2525x =

00

650650

0

Find the probabilityFind the probability

P(striped, P(striped, striped)striped)

There are 10 There are 10 marbles in the marbles in the bag:bag: 5 striped5 striped 5 solid5 solid

Do not put the Do not put the first marble back.first marble back.

55

1010

44

99x =

2020

9090

22

99

The num. changed to a 4 because you pulled out a

striped marble. Now the bag is missing one.

The denominator changed to a 9

because we have one less marble in the bag since we

did not put it back!