Independent Events

1. Independent Events Two events A and B are independent if the fact that A occurs does not change the probability of B occurring

2. Dependent Events When the outcome of the first event changes the probability of the second event occurring

3. Conditional Probability The probability that event B occurs knowing that event A already occurred Symbol: P(B|A) (read “Probability of B given A”)

4. P(A and B) for Independent Events When two events are independent, the probability of both occurring is: P(A and B) = P(A) * P(B)

5. P(A and B) for dependent Events When two events are dependent, the probability of both occurring is: P(A and B) = P(A) * P(B|A)

6. Example: P(A and B) for dependent Events Using a deck of cards and selecting two cards, find the probability that both are spades Event A = Spade Event B = Spade Does event A have any affect on event B occurring? Yes, they are dependent

7. Example: P(A and B) for dependent Events Using a deck of cards and selecting two cards, find the probability that both are spades Event A = Spade Event B = Spade P(A and B) = P(A) * P(B) = P(spade) * P(spade) = 13/52 * 12/51 = 1/17

8. Example: P(A and B) for dependent Events Selecting names from a hat with 12 names. What is the probability of drawing Annie and Justin without replacing? Event A = Annie Event B = Justin Does event A have any affect on event B occurring? Yes, they are dependent

9. Example: P(A and B) for dependent Events Selecting names from a hat with 12 names. What is the probability of drawing Annie and Justin without replacing? Event A = Annie Event B = Justin P(A and B) = P(A) * P(B) = P(Annie) * P(Justin) = 1/12 * 1/11 = 1/132

10. Example: P(A and B) for Independent Events Selecting names from a hat with 12 names. What is the probability of drawing Annie and Justin replacing each time? Event A = Annie Event B = Justin Does event A have any affect on event B occurring? No, they are independent

11. Example: P(A and B) for Independent Events Selecting names from a hat with 12 names. What is the probability of drawing Annie and Justin replacing each time? Event A = Annie Event B = Justin P(A and B) = P(A) * P(B) = P(Annie) * P(Justin) = 1/12 * 1/12 = 1/144

12. Example: P(A and B) for Independent Events What is the probability of rolling a die and Getting a number greater than 4, then flipping a coin and getting tails? Event A = Greater than 4 Event B = Tails Does event A have any affect on event B occurring? No, they are independent

13. Example: P(A and B) for Independent Events What is the probability of rolling a die and Getting a number greater than 4, then flipping a coin and getting tails? Event A = Greater than 4 Event B = Tails P(A and B) = P(A) * P(B) = P(<4) * P(Tails) = 2/6 * 1/2 = 1/6

14. Example similar to homework Dealing with extremely large populations: If 6% of Americans use an umbrella, find the probability that 3 randomly selected Americans all use an umbrella. Probability that 1st person uses an umbrella: P(umbrella) = 0.06 Even though this person is taken out of the population after being surveyed, would that change the probability enough to change our data? No, since it is only one person out of an extremely large population, you wouldn’t notice a change in the probability.

15. Example similar to homework Dealing with extremely large populations: If 6% of Americans use an umbrella, find the probability that 3 randomly selected Americans all use an umbrella. Probability that 1st person uses an umbrella: P(umbrella) = 0.06 Probability that 2nd person uses an umbrella: P(umbrella) = 0.06 Probability that 3rd person uses an umbrella: P(umbrella) = 0.06 P(all 3 use umbrella) = 0.06 * 0.06 * 0.06 = 0.0002

16. Independent/Dependent: p201 # 1, 4 – 16 even (OMIT #10)