Independent and Dependent Events Goal: identify the difference between a dependent and an independent event.

Independent Events Frequently you will determine the probability of multiple events. You’ll need to decide if the events are independent events or dependent events. Events are independent if the occurrence of one event does not affect the probability of the other. If you roll a single die twice, the results of the first roll do not affect the probability of the other.

Find the P(3, 3). The first time you roll the die, the probability of rolling a 3 is 1 6 . When you roll the die again, the probability of rolling a 3 again is still 1 6 . These two events are independent. To find the probability of rolling a 3 and then another 3 is 1 6 • 1 6 or 1 36

Other examples of independent events… Flipping a coin more than once Spinning a spinner more than once Choosing items off a menu (assuming the kitchen has an unlimited supply of all menu items) Picking a marble out of a bag, replacing it, then picking another marble Drawing a card, replacing it, then drawing another card Choosing a shirt and pants to wear from a group of choices

Dependent Events Dependent Events – the occurrence of one event does affect the probability of the other. In bag are 10 marbles, 4 green, 4 red, and 2 yellow. Danny will draw a marble and will not replace it. Then Austin will draw a marble. What is the probability that both will draw a yellow marble, P(yellow, yellow)?

Other examples of dependent events… Drawing a card without replacement then drawing another card Sitting center courtside at a Suns game Choosing items off a dessert cart (no more in the kitchen) Finding a shopping cart at the grocery store Choosing a number 1 – 10 to win a prize Picking out a marble, not replacing it and then picking out another marble Finding a seat at a certain table in a classroom Choosing a chocolate out of a box of chocolates

Now lets practice identifying independent and dependent events…..

Probability of Independent Compound Events 1. Calculate the probability of the first event. 2. Calculate the probability that the second event would occur. 3. Multiply the probabilities. If A and B are independent events, then P(A and B) = P(A) = P(B)

Finding the Probability of Independent Events 1. An experiment consists of spinning the spinner 3 times. For each spin, all outcomes are equally likely. What is the probability of spinning a 5 all 3 times? P(5, 5, 5)

a. What is the probability of spinning an odd number all 3 times?

3. Three separate boxes each have one blue marble and one green marble 3. Three separate boxes each have one blue marble and one green marble. One marble is chosen from each box. a. What is the probability of choosing a blue marble from each box? b. What is the probability of choosing a blue marble, then a green marble and then a blue marble?

Dependent Events 1. Calculate the probability of the first event. 2. Calculate the probability that the second event would occur assuming the first event had already occurred. 3. Multiply the probabilities. If A and B are dependent events, then P(A and B) = P(A) • P(B after A).

1. Suppose you draw 2 marbles without replacement from a bag that contains 3 purple and 3 orange marbles. On the first draw: P(purple) = The P(purple, purple) =

2. The letters in the word dependent are placed in a box. A. If two letters are chosen at random without replacement, what is the probability that they will both be consonants? B. If two letters are chosen at random without replacement, what is the probability that they will both be vowels?

3. A bag contains 5 chocolate chip, 3 peanut butter, 4 oatmeal, and 4 sugar cookies. A. If Tia randomly selects 2 cookies, what is the probability that they will both be sugar cookies? B. If Vanessa randomly chooses 2 cookies, what is the probability that the first will be chocolate chip and the second a sugar cookie?

4. A drawer contains 10 black socks and 6 blue socks. A. If 2 socks are chosen at random, find P(black pair). B. If 2 socks are chosen at random, find P(one black, one blue).

1. An experiment consists of spinning each spinner once. A. Find the probability that the first spinner lands on 5 and the second lands on 3. B. Find the probability that both spinners land on an odd number.

C. Find the probability that the first spinner lands on an even number and the second lands on an odd number.

2. A box contains 5 red marbles, 3 blue marbles, and 7 white marbles. A. Find P(red then blue) if a marble is selected, then a second is selected without replacing the first marble. B. Find P(red then blue) if a marble is selected and replaced, then a second marble is selected.

3. A bucket contains 5 yellow and 7 red balls 3. A bucket contains 5 yellow and 7 red balls. If 2 balls are selected randomly without replacement, what is the probability that they will both be yellow?