Probability Chapter 11 1

Odds and Mathematical Expectation Section 11.6 2

Probability to Odds If P(E) is the probability of an event E occurring, then NOTE: The odds against E can also be found by reversing the ratio representing the odds in favor of E. Also odds is not probability and probability is not odds but one can be found from the other. 3

Examples Find the odds in favor of obtaining a 2 in one roll of a single die. an ace when drawing 1 card from an ordinary deck of 52 cards. at least 1 head when an ordinary coin is tossed 3 times. 4

Solutions 5

Examples Find the odds against obtaining a 2 every time in three rolls of a single die. exactly 2 tails when an ordinary coin is tossed 3 times. one of the face cards when drawing 1 card from an ordinary deck of 52 cards. 6

Solutions 7

Examples According to a survey, the probability of being the victim in a serious crime in your lifetime is 1/20. Find the odds in favor of this event occurring. the odds against this event occurring. 8

Odds to Probability If the odds in favor of an event E are a to b, then the probability of the event is given by If the odds in favor of an event E are a to b, then the probability of the event not happening is given by 9

Examples The odds in favor of having complications during surgery in June are 1 to 4. What is the probability that this event will occur? What are the odds against this event occurring? What is the probability that this event will not occur? 10

Solutions 11

Expected Value Expected value is a mathematical way to use probabilities to determine what to expect in various situations over the long run. It is used to weigh the risks versus the benefits of alternatives in business ventures, and indicate to a player of any game of chance what will happen if the game is played a large number of times. 12

Expected Value If the k possible outcomes of an experiment are assigned the values a1, a2, …, ak and they occur with probabilities p1, p2, …, pk, respectively, then the expected value of the experiment is given by 13

Examples A coin is tossed twice. If exactly 1 head comes up, we receive $2, and if 2 tails come up, we receive $4; otherwise, we lose $10. What is the expected value of this game? Possible outcomes are HH, HT, TH, and TT. 14

Examples In a roulette game, the wheel has 38 compartments, 2 of which, the 0 and 00, and the rest are numbered 1 through 36. You can either win $17 if 0 or 00 comes up, or lose $1.00 if any other number comes up. What is the expected value of this game? 15

Examples Suppose you have the choice of selling hot dogs at two stadium locations. At stadium A you can sell 100 hot dogs for $4 each, or if you lower the price and move to stadium B, you can sell 300 hot dogs at $3 each. The probability of being assigned to A is .55, and to B is .45. Find the expected value for A B Which location would you choose? 16

Solutions 17

Examples A store specializing in mountain bikes is to open in one of two malls, described as follows: 1st Mall: Profit if store is successful: $300,000 Loss if it is unsuccessful: $100,000 Probability of success: ½ 2nd Mall: Profit if store is successful: $200,000 Loss if it is unsuccessful: $60,000 Probability of success: ¾ Which mall should be chosen in order to maximize the expected profit? 18

Solutions Note: This is ½ because 1 – ½ = ½ not because both parentheses are the same. Note: 1 – ¾ = ¼ . 19 END