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math

1. 1. By Jaclyn Kogut Independent and Dependent Events 11-3
2. 2. Objectives  <ul><li>Determine whether events are independent or dependent. </li></ul><ul><li>Find the probability of independent and dependent events. </li></ul>
3. 3. Vocabulary <ul><li>Independent events: </li></ul><ul><li>Dependent events: </li></ul><ul><li>The occurrence of one </li></ul><ul><li>event that does not </li></ul><ul><li>affect the probability of </li></ul><ul><li>the other </li></ul><ul><li>The occurrence of one </li></ul><ul><li>event that affects the </li></ul><ul><li>probability of the other </li></ul><ul><li>Conditional Probability </li></ul>
4. 4. Conditional Probability <ul><li>When finding the probability of dependent events, you use conditional probability. </li></ul><ul><li>P(B  A) = the probability of event B , given that event A has already occurred </li></ul>
5. 5. Probability of Independent Events <ul><li>If A and B are independent events, then </li></ul><ul><li>P(A and B) = P(A)  P(B) </li></ul>
6. 6. Example 1: Finding the Probability of Independent Events <ul><li>There is a ten-sided cube: three sides are colored green, two sides are colored red, and five sides are colored blue </li></ul><ul><li>Find the probability of rolling a green, then red, then blue </li></ul>
7. 7. Example 1: Finding the Probability of Independent Events <ul><li>P(green, then red, and then blue) = P(green)  P(red)  P(blue) </li></ul>
8. 8. Example 1: Finding the Probability of Independent Events Step 1: Find the probability of rolling one green side P(green) = 3/10 Step 2: Find the probability of rolling one red side P(red) = 2/10 = 1/5 Step 3: Find the probability of rolling one blue side P(blue) = 5/10 = 1/2
9. 9. Example 1: Finding the Probability of Independent Events <ul><li>Step 4: Multiply each separate probability by each other </li></ul><ul><li>P(green, then red, and then blue) = P(green)  P(red)  P(blue) </li></ul><ul><li> = 3/10  1/5  1/2 </li></ul><ul><li> = 3/100 </li></ul><ul><li>= 0.03 probability </li></ul>
10. 10. Probability of Dependent Events <ul><li>If A and B are dependent events, then </li></ul><ul><li>P(A and B) = P(A)  P(B  A), where </li></ul><ul><li>P(B  A) is the probability of B, given that A has already occurred </li></ul>
11. 11. Example 2: Finding the Probability of Dependent Events <ul><li>Two number cubes are rolled - one blue and one white. Explain why the events are dependent and find the probability. </li></ul><ul><li>The blue cube must show a 5 and the sum of the two cubes must be greater than 8 </li></ul>
12. 12. Example 2: Finding the Probability of Dependent Events <ul><li>P(blue 5 and sum  8) = P(blue 5)  P(sum  8  blue 5) </li></ul>
13. 13. Example 2: Finding the Probability of Dependent Events <ul><li>Step 1: Find the probability of blue rolling a 3 </li></ul><ul><li>P(blue 5) = 2/12 = 1/6 </li></ul><ul><li>Step 2: Find the probability of the white cube rolling a number that will make the sum greater than 8 </li></ul><ul><li>P(sum  8  blue 5) = 4/6 = 2/3 </li></ul>
14. 14. Example 2: Finding the Probability of Dependent Events <ul><li>Step 3: Multiply each probability together to get the final probability </li></ul><ul><li>P(blue 5 and sum  8) = P(blue 5)  P(sum  8  blue 5) </li></ul><ul><li>= 1/6  2/3 </li></ul><ul><li>= 2/18 = 1/9 </li></ul><ul><li>= .11111111111 probability </li></ul>
15. 15. Determine if the event is independent or dependent: <ul><li>Selecting a spade when the first card is taken out of the deck </li></ul><ul><li>Drawing a queen, putting it back, and then drawing a king </li></ul><ul><li>Eating all the red skittles and taking out a green skittle </li></ul><ul><li>Taking out all the green skittles, putting 5 back, eating a purple, and finding a orange skittle </li></ul>
16. 16. THE END