1. Thinking Mathematically Events Involving And; Conditional Probability

2. Independent Events Two events are independent events if the occurrence of either of them has no effect on the probability of the other.

3. And Probabilities with Independent Events If A and B are independent events, then P(A and B) = P(A)P(B)

4. Example Independent Events on a Roulette Wheel A roulette wheel has 38 numbered slots (1 through 36, 0, and 00). Of the 38 compartments, 18 are black, 18 are red, and two are green. A play has the dealer spin the wheel and a small ball in opposite directions. As the ball slows to a stop, it can land with equal probability on any one of the 38 numbered slots. Find the probability of red occurring on two consecutive plays.

5. Solution The wheel has 38 equally likely outcomes and 18 are red. Thus the probability of red occurring on a play is 18/38, or 9/19. The result that occurs on each play is independent of all previous results. Thus, P(red and red) = P(red)P(red) = 9/199/19 = 81/361

6. Example Hurricanes and Probability If the probability that South Florida will be hit by a hurricane in any single year is 5/19, what is the probability that South Florida will be hit by a hurricane in three consecutive years?

7. Solution The probability that South Florida will be hit by a hurricane in three consecutive years is: P(hurricane and hurricane and hurricane) = P(hurricane)P(hurricane)P(hurricane) = 5/195/195/19 = 125/6859

8. Dependent Events Two events are dependent events if the occurrence of one of them has an effect on the probability of the other.

9. And Probabilities with Dependent Events If A and B are dependent events, then P(A and B) = P(A)P(B given that A has occurred)

10. Example Dependent Events with Your Cousins Good news: You won a free trip to Madrid and can take two people with you, all expenses paid. Bad news: Ten of your cousins have appeared out of nowhere and are begging you to take them. You write each cousin’s name on a card, place the card in a hat, and select one name. Then you select a second name without replacing the first card. If three of your ten cousins speak Spanish, find the probability of selecting two Spanish-speaking cousins.

11. Solution Because P(A and B) = P(A)P(B given that A has occurred), then P(two Spanish-speaking cousins) = P(speaks Spanish)P(speaks Spanish given that a Spanish-speaking cousin was selected first) = 3/10 2/9 = 6/90 = 1/15. The probability of selecting two Spanish-speaking cousins is 1/15.

12. Example An And Probability with Three Dependent Events Three people are randomly selected, one person at a time, from five freshman, two sophomores, and four juniors. Find the probability that the first two people selected are freshman and the third is a junior.

13. Solution P(first two are freshman and the third is a junior) = P(freshman)P(freshman given that a freshman was selected first)P(junior given that a freshman was select first and second) = 5/11 4/10 4/9 = 8/99 The probability that the first two are freshman and the third is a junior is 8/99.

14. Conditional Probability The conditional probability of B, given A, written P(B|A), is the probability that event B occurs computed on the assumption that event A occurs.

15. Example Computing Conditional Probability A letter is randomly selected from the letters of the English alphabet. Find the probability of selecting a vowel, given that the outcome is a letter that precedes h.

16. Solution Because we are given the condition that the outcome is a letter that precedes h, the set of all possible outcomes is {a, b, c, d, e, f, g} There are seven possible outcomes. We can select a vowel from this set in one of two ways: a or e. Therefore, the probability of selecting a vowel, given that the outcome is a letter than precedes h, is 2/7. P(vowel | outcome precedes h) = 2/7

17. Applying Conditional Probability to Real-World Data P(B|A) = observed number of times B and A occur together observed number of times A occurs

18. Example Conditional Probabilities with Real-World Data The table shows the number of active-duty personnel in the U.S. military in 2000. If one person is randomly selected from the U.S. military, find the probability that person is male if the person is in the Marine Corps.

19. Solution We find the probability that the selected person is a male, given that we are selecting from the Marines. P(Male|Marine Corps) = Observed number of people who are males and Marines Observed number of people in the Marines =(161,571)/(161,571+10,130) = 161,571/171,701 = 0.941

20. Thinking Mathematically Events Involving And; Conditional Probability