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Thinking Mathematically

Thinking Mathematically. Events Involving Not and Or; Odds. The Probability of an Event Not Occurring. The probability that an event E will not occur is equal to one minus the probability that it will occur. P(not E) = 1 - P(E). Example The Probability of an Event Not Occurring.

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Thinking Mathematically

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  1. Thinking Mathematically Events Involving Not and Or; Odds

  2. The Probability of an Event Not Occurring The probability that an event E will not occur is equal to one minus the probability that it will occur. P(not E) = 1 - P(E)

  3. Example The Probability of an Event Not Occurring If you are dealt one card from a standard 52-card deck, the probability that you are not dealt a queen is equal to one minus the probability that you are dealt a queen. P(not queen) = 1 - P(queen). There are four queens in a deck of 52 cards. The probability of being dealt a queen is 4/52 = 1/13. Thus, P(not queen) = 1 - 1/13 = 12/13.

  4. Example The Probability of Not Winning the LOTTO We have seen the probability of winning Florida’s LOTTO with one ticket is 1/22,957,480. Hence, P(not winning) = 1 - P(winning) = 1 - 1/22,957,480 = 22,957,479/22,957,480 = .99999996

  5. Mutually Exclusive Events If it is impossible for events A and B to occur simultaneously, the events are said to be mutually exclusive. If A and B are mutually exclusive events, thenP(A or B) = P(A) + P(B).

  6. Example The Probability of Either of Two Mutually Exclusive Events Occurring If one card is randomly selected from a deck of cards, what is the probability of selecting a king or a queen?

  7. Solution We find the probability that either of these mutually exclusive events will occur by adding their individual probabilities. P(king or queen) = P(king) + P(queen) = 4/52 + 4/52 = 8/52 = 2/13

  8. Or Probabilities with Events That Are Not Mutually Exclusive If A and B are not mutually exclusive events, then P(A or B) = P(A) + P(B) - P(A and B)

  9. Example An Or Probability with Events That Are Not Mutually Exclusive In a group of 25 baboons, 18 enjoy picking fleas off of their neighbors, 16 enjoy screeching wildly, while 10 enjoy picking fleas off of their neighbors and screeching wildly. If one baboon is selected at random from the group, find the probability that it enjoys picking fleas off its neighbor or screeching wildly.

  10. Solution It is possible for a baboon in the group to enjoy picking its neighbor’s fleas and screeching wildly. These events are not mutually exclusive. P(picking fleas or screeching wildly) = P(picking fleas)+P(screeching wildly) - P(picking fleas and screeching wildly) = 18/25 + 16/25 - 10/25 = 24/25

  11. Example An Or Probability with Events That Are Not Mutually Exclusive A group of people is comprised of 15 U.S. men, 20 U.S. women, 10 Canadian men, and 5 Canadian women. If a person is selected at random from the group, find the probability that the selected person is a man or a Canadian.

  12. Solution The group is comprised of 15+20+10+5 = 50 people. It is possible to select a man who is Canadian, so these events are not mutually exclusive. P(man or Canadian) = P(man) + P(Canadian) - P(Canadian man) = 25/50 + 15/50 - 10/50 = 30/50 = 3/5

  13. Odds If there are a outcomes that are favorable to event E and b outcomes that are unfavorable to event E, then • The odds in favor of E are a to b, written a:b. • The odds against E are b to a, written b:a.

  14. Example Stating the Odds You roll a single, six-sided die. The odds in favor of getting 2 are the number of favorable outcomes, 1, to the number of unfavorable outcomes, 5. Thus, the odds in favor of getting 2 are 1 to 5, written 1:5. The odds against getting 2 are the number of unfavorable outcomes, 5, to the number of favorable outcomes, 1. Thus, the odds against getting 2 are 5 to 1, written 5:1.

  15. Example Stating the Odds The winner of a raffle will receive a new sports utility vehicle. If 500 raffle tickets were sold and you purchased ten tickets, what are the odds against your winning the car?

  16. Solution The odds against your winning the car are: The number of outcomes that are unfavorable for winning the car, 490, to the number of outcomes that are favorable for winning the car, 10. Thus, the odds against your winning the car are 490 to 10, or 49 to 1, written 49:1.

  17. Finding Probabilities from Odds If the odds in favor of an event E are a to b, then the probability of the event is given by P(E) = a a+b

  18. Example From Odds to Probability The odds in favor of a particular horse winning a race are 2 to 5. What is the probability that this horse will win the race?

  19. Solution Because the odds in favor, a to b, means a probability of a / (a+b), then odds in favor, 2 to 5, means a probability of 2 / (2+5) = 2/7. The probability that this horse will win the race is 2/7.

  20. Thinking Mathematically Events Involving Not and Or; Odds

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