LEARNING GOAL The student will understand how to calculate the probability of an event.

OBJECTIVE #2 The student will be able to calculate the probability of independent and dependent events with 80% accuracy.

VOCABULARY Independent Event: when the outcome of an event does not affect the outcome of another event Example: when rolling a die twice, the probability of getting a 3 on the second roll is the same as getting a 3 on the first roll Dependent Event: when the outcome of an event changes the probability of the outcome of another event Example: if we have a bag of 10 candies, 5 red and 5 orange, the probability of randomly choosing and eating a red candy on the second draw is different than randomly choosing and eating a red candy on the first draw

INDEPENDENT EVENTS If A and B are independent events, then the probability of A and B is P(A) x P(B) Example #1: What is the probability of getting two heads when a quarter and a nickel are flipped? Event A is the flip of the quarter, and 𝑃 𝐴 = 1 2 Event B is the flip of the nickel, and 𝑃 𝐵 = 1 2 𝑃 𝑡𝑤𝑜 ℎ𝑒𝑎𝑑𝑠 = 1 2 × 1 2 = 1 4

INDEPENDENT EVENTS Example #2: What is the probability of choosing 2 Aces from a standard deck of 52 cards when the first card is put back into the deck before the second card is drawn? Event A is choosing the first ace, and 𝑃 𝐴 = 4 52 = 1 13 Event B is choosing the next ace, and 𝑃 𝐵 = 4 52 = 1 13 𝑃 𝑡𝑤𝑜 𝑎𝑐𝑒𝑠, 𝑤𝑖𝑡ℎ 𝑟𝑒𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 = 1 13 × 1 13 = 1 169

DEPENDENT EVENTS If A and B are dependent events, then the probability of A and B is P(A) x P(B*) where P(B*) is the probability that B happens given that A already happened. Example #1: In a class of 12 boys and 16 girls, what is the probability that two students randomly selected by the teacher are two boys? Event A is choosing the first boy, and 𝑃 𝐴 = 12 28 = 3 7 Event B is choosing the second boy, and 𝑃 𝐵 ∗ = 11 27 𝑃 𝑡𝑤𝑜 𝑏𝑜𝑦𝑠 𝑎𝑟𝑒 𝑐ℎ𝑜𝑠𝑒𝑛 = 3 7 × 11 27 = 33 189 = 11 63

INDEPENDENT EVENTS Example #2: What is the probability of choosing 2 Aces from a standard deck of 52 cards when the first card is not put back into the deck before the second card is drawn? Event A is choosing the first ace, and 𝑃 𝐴 = 4 52 = 1 13 Event B is choosing the next ace, and 𝑃 𝐵 ∗ = 3 51 = 1 17 𝑃 𝑡𝑤𝑜 𝑎𝑐𝑒𝑠 𝑤𝑖𝑡ℎ𝑜𝑢𝑡 𝑟𝑒𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 = 1 13 × 1 17 = 1 221

GUIDED PRACTICE Are the following scenarios independent or dependent events? A student spins a spinner and rolls a die A student picks a raffle ticket and then picks a second ticket without replacing the first one A student picks a raffle ticket and then picks a second ticket after the first one was replaced Find the probability of drawing a 1, 2, or 3 from 9 cards numbered 1-9, replacing the card and drawing a 7, 8, or 9.

GUIDED PRACTICE Find the probability of drawing a black checker from a bag of 6 black and 4 red checkers, replacing it, and drawing another black checker. Are the two draws independent or dependent? Assuming the first checker was not replaced, what is the probability of drawing two black checkers?