Independent and Dependent Events Goal: To find the probability of two independent or two dependent events.

Independent Events Events for which the occurrence of the FIRST DOES NOT effect that of the SECOND because: Events for which the occurrence of the FIRST DOES NOT effect that of the SECOND because: 1. The two events are unrelated OR 2. You repeat an event with an item whose numbers will not change (ex: spinners or dice) OR 3. You repeat the same activity, but you REPLACE the item that was removed.

The 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 #1

Independent Events #2 You repeat an event with an item whose numbers will not change (e.g.: spinners, dice, flipping a coin, etc.)

Independent Events #3 You repeat the same activity by pulling something out, but you REPLACE the item that was removed.

Probability of Independent Events When you have two independent events: When you have two independent events: Find the probability of the first event Find the probability of the first event Find the probability of the second event Find the probability of the second event Multiply the probabilities together Multiply the probabilities together Reduce your answer Reduce your answer For independent events A and B, the probability of both events occurring is found by multiplying the probabilities of the events. For independent events A and B, the probability of both events occurring is found by multiplying the probabilities of the events. P(A and B) = P(A) P(B) P(A and B) = P(A) P(B)

An Independent Event There are 10 socks in a basket: 5 blue, 3 yellow and 2 red. What is the probability of choosing a yellow sock and a red sock if the sock is replaced after the first event? is replaced P(yellow, red) Yellow: Replace the yellow sock. Red: =15 x350

Probability of Independent Events A bag containing 4 green marbles, 6 red marbles, and 2 white marbles. Three marbles are drawn at random with replacement. With replacement means that after a marble is drawn, it is replaced before the next one is drawn. Find each probability. 1. P(green, red, white) 2. P(green, white, white) 3. P(all three red) 4. P(not green, not red, not white)

Dependent Events Events for which the occurrence of the FIRST AFFECTS that of the SECOND Events for which the occurrence of the FIRST AFFECTS that of the SECOND Look for the following 2 clues to determine if an event is dependent Look for the following 2 clues to determine if an event is dependent You remove something after the first event and DO NOT REPLACE it. You remove something after the first event and DO NOT REPLACE it. You ask yourself the question, “ What happens second? ” and your answer is “ It depends... ” You ask yourself the question, “ What happens second? ” and your answer is “ It depends... ”

Probability of Dependent Events When you have two dependent events: When you have two dependent events: Find the probability of the first event Find the probability of the first event Figure out what has changed Figure out what has changed Numerator and/or denominator Numerator and/or denominator Figure out the probability of the second event Figure out the probability of the second event Multiply the 2 probabilities together Multiply the 2 probabilities together Reduce your answer Reduce your answer For dependent events A and B, the probability of both events occurring is found by multiplying the probabilities of the events. For dependent events A and B, the probability of both events occurring is found by multiplying the probabilities of the events. P(A and B) = P(A) P(B) P(A and B) = P(A) P(B)

A Dependent Event There are 10 socks in a basket: 5 blue, 3 yellow and 2 red. What is the probability of choosing a yellow sock and a red sock if the sock is NOT replaced after the first event? is NOT replaced P(yellow, red) Yellow: DO NOT replace the yellow sock. Red: x = Notice that the denominator changed

Probability of Dependent Events At a meeting there are 6 juniors and 12 seniors. Four people are selected at random, one at a time for a committee. Find each probability. 1. P(all juniors) 2. P(junior, junior, senior, senior) 3. P(senior, senior, senior, junior) 4. P(all senior)

TEST YOURSELF Are these dependent or independent events?

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

Are these dependent or independent events? You pick the letter Q from a bag containing all the letters of the alphabet. You do not put the Q back in the bag before you pick another tile. You pick the letter Q from a bag containing all the letters of the alphabet. You do not put the Q back in the bag before you pick another tile. Dependent Event

Are these dependent or independent events? You have a bag of marbles: 3 blue, 5 white, and 12 red. You choose one marble out of the bag, look at it then put it back. Then you choose another marble. You have a bag of marbles: 3 blue, 5 white, and 12 red. You choose one marble out of the bag, look at it then put it back. Then you choose another marble. Independent Event

Are these dependent or independent events? You pick the letter W from a bag containing all the letters of the alphabet. You put the W back in the bag and pick a second time. You pick the letter W from a bag containing all the letters of the alphabet. You put the W back in the bag and pick a second time. Independent Event

Are these dependent or independent events? You have a basket of socks. You need to find the probability of pulling out a black sock and its matching black sock without putting the first sock back. Dependent Event

Find the probability P(jack, green) P(jack, green) x=

Find the probability P(6, not 5) P(6, not 5) x=536

Find the probability P(Q, Q) P(Q, Q) All the letters of the alphabet are in the bag 1 time All the letters of the alphabet are in the bag 1 time Do not replace the letter Do not replace the letter x=0650 0

Find the probability P(striped, striped) P(striped, striped) There are 10 marbles in the bag: There are 10 marbles in the bag: 5 striped 5 striped 5 solid 5 solid Do not put the first marble back. Do not put the first marble back x= 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!