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COMPOSITE EVENTS

COMPOSITE EVENTS. The probability that it rains is 70%. The probability that it does NOT rain is. 30%. Instinct tells us that for any event E , the probability that E happens + the probability that E does NOT happen = 100%.

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COMPOSITE EVENTS

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  1. COMPOSITE EVENTS

  2. The probability that it rains is70% The probability that it does NOT rain is 30% Instinct tells us that for any event E, the probability that E happens + the probability that E does NOThappen = 100% P(E) + P(E’) = 100% = 100/100 = 1

  3. H H T H H T T H H T T H T T Flip 3 coins: 2 x 2 x 2 = 8 possible outcomes

  4. H H T H H T T H H T T H T T Flip 3 coins: 2 x 2 x 2 = 8 possible outcomes NOT ALL TAILS The probability that at least one lands heads up = 7/8 ALL TAILS The probability that all land tails up = 1/8

  5. FLIP 10 COINS What is the probability that at least one lands heads up? 10 The number of possible outcomes = 2 = 1024 P( all tails ) = 1/1024 P( NOT all tails ) = P(at least one heads up ) = 1023/1024

  6. Favorable outcomes Roll 4 dice 6 x 6 x 6 x 6 = 1296 possible outcomes What is the probability that at least two land on the same number? Some of the possible outcomes: 231 6 5251 4443 1635 3421 1262 . .

  7. Roll 4 dice It would be very difficult to count the number of ways in which at least two land on the same number. Instead let’s try to count the number of ways in which it does NOT happen that at least two are the same ….. That is They ALL land on DIFFERENT numbers # ways blue die can land: 6 # ways red die can land: x5 # ways yellow die can land: x4 # ways green die can land: x3 = 360

  8. # ways They ALL land on DIFFERENT numbers # ways blue die can land: 6 # ways red die can land: x5 # ways yellow die can land: x4 # ways green die can land: x3 = 360 Roll 4 dice 6 x 6 x 6 x 6 = 1296 possible outcomes P( all different ) = 360/1296 P( NOT all different ) = 1 - 360/1296

  9. P( NOT all different ) = 1 - 360/1296 1296/1296 - 360/1296 936/1296

  10. The probability that you get an A in French is 1/10 The probability that you get an B in French is 3/10 What is the probability that you get an A ORa B in French? Instinct tells us to add when we see the word OR 1/10 + 3/10 = 4/10 = P( A OR B ) In some cases, there is more to consider.

  11. shuffle this deck of 10 cards and draw one at random. What is the probability that it is a white card OR a King ?

  12. P ( white card ) = P( W ) = 5/10 P ( King ) = P( K ) = 2/10 P ( white OR King ) = 6/10

  13. P ( white card ) = P( W ) = 5/10 P ( King ) = P( K ) = 2/10 P ( white OR King ) = 6/10

  14. # possible outcomes = # ways of drawing 2 red cards = # ways of drawing 2 kings = Draw 2 cards without replacement from a standard deck of 52 cards. What is the probability that both are red OR both are kings? P ( both red ) = P( R ) = 325/1326 P ( both kings ) = P( K ) = 6/1326

  15. = 1/1326 P ( both red ) = P( R ) = 325/1326 P ( both kings ) = P( K ) = 6/1326 Both are red AND both are kings = only 1 of the 1326 possible hands satisfies this condition

  16. P ( both red ) = P( R ) = 325/1326 P ( both kings ) = P( K ) = 6/1326 = 1/1326

  17. There are 10 women in a study.

  18. S 3 of them smoke There are 10 women in a study.

  19. S H 3 of them smoke 3 develop heart disease There are 10 women in a study.

  20. S H The probability that a woman in this sample develops heart disease = P(H)= 3/10 3 develop heart disease There are 10 women in a study.

  21. S The probability that she gets heart disease IF she smokes = H

  22. S The probability that she gets heart disease IF she smokes = 2/3

  23. S H P(H) = 3/10 = 30% P(H IF S) P(H/S) = 2/3 = 66.7% Based on this data, we might conclude that a woman is more than twice as likely to develop heart disease IF she smokes.

  24. S H P(H IF S) P(H/S) = 2/3 = 66.7%

  25. S H P(H IF S) P(H/S) = 2/3 = 66.7%

  26. The probability that B happens IF A happens = the probability that A happens AND B happens the probability that A happens We rewrite this formula to obtain a formula for measuring the probability that A happens AND B happens

  27. Multiply both sides of the equation by P(A):

  28. R C There is a 40% chance that my car won’t start IF it rains. There is a 70% chance of rain. What is the probability that it rains AND my car won’t start. (.40) (.70) = .28

  29. P(both blue) = P(first is blue AND second is blue) A jar contains 2 red and 3 blue marbles. Draw two without replacement. What is the probability they are both blue?

  30. NOT OR IF AND

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