Advanced Algebra Notes Section 10.5: Find Probabilities of Independent and Dependent Events Two events are independent if the occurrence of one has no effect on the occurrence of the other. For instance, if a coin is tossed twice, the outcome of the first toss (heads or tails) has no effect on the outcome of the second toss. Probability of Independent Events If A and B are independent events, then the probability that both A and B occur is: π π΄ πππ π΅ = π(π΄) ∗ π(π΅) More generally, the probability that n independent events occur is the product of the n probabilities of the individual events. Example 1 In a BMX meet, each heat consists of 8 competitors who are randomly assigned lanes from 1 to 8. What is the probability that a racer will draw lane 8 in the 3 heats in which the racer participates? A is first heat B is second heat C is third heat π π΄ πππ π΅ πππ πΆ = π π΄ ∗ π π΅ ∗ π(πΆ) 1 1 1 = ∗ ∗ 8 8 8 1 = 15 ≈ .00195 Example 2 While you are riding to school, your portable CD player randomly plays 4 different songs from a CD with 16 songs on it. What is the probability that you will hear your favorite song on the CD at least once during the week (5 days)? π πππ‘ βππππππ π πππ = =1− Hearing your song on Monday and Tuesday are independent π π»ππππππ ππππ = 1 − [π πππ‘ βππππππ π‘βπ π πππ ] 5 ≈ 0.763 5 Two events A and B are dependent events if the occurrence of one affects the occurrence of the other. The probability that B will occur given that A has conditional events occurred is called the of B given A is written as π π΅ π΄ Probability of Dependent Events If A and B are dependent events, then the probability that both A and B occur is: π π΄ πππ π΅ = π(π΄) ∗ π(π΅|π΄) Example 4 The table shows the numbers of tropical cyclones that formed during the hurricane seasons from 1988 to 2004. Use the table to estimate a. The probability that a future tropical cyclone is a hurricane # ππ π»π’ππππππππ π π»π’πππππππ = πππ‘ππ πΆπ¦ππππππ b. = 760 1575 ≈ .483 The probability that a future tropical cyclone in the Northern Hemisphere is a hurricane. # ππ π»π’ππππππππ ππ π. π»ππππ πβπππ = πππ‘ππ πΆπ¦ππππππ ππ π. π»ππππ πβπππ 545 = 1142 ≈ .477 Example 5 You randomly select two cards from a standard deck of 52 cards. That is the probability that the first card is not a heart and the second is a heart if: a. You replace the first card before selecting the second π π΄ πππ π΅ = π π΄ ∗ π(π΅) b. 39 13 = ∗ 52 52 3 = 16 ≈ .188 You do not replace the first card? π π΄ πππ π΅ = π π΄ ∗ π π΅ π΄ 39 13 = ∗ 52 51 13 = 68 ≈ .191 Example 6 You and two friends go to the same store at different times to buy costumes for a costume party. There are 15 different costumes at the store, and the store has at least 3 duplicates of each costume. What is the probability that you each choose different costumes? A= Costume B= Different Costume Chosen π π΄, π΅ πππ πΆ = π π΄ ∗ π π΅ π΄ ∗ π πΆ π΄ πππ π΅ 15 14 13 = ∗ ∗ 15 15 15 182 = 225 ≈ .809 C=Friend Choses a Third Costume