Statistical Reasoning for everyday life Intro to Probability and Statistics Mr. Spering – Room 113

6.3 What to expect in the long run… Find the probability.  What is the probability of rolling a number less than 8 on a number cube?  1 or 100%  A bag contains 30 red marbles, 50 blue marbles, and 20 white marbles. You pick one marble from the bag. Find P (picking a blue).  0.5 or 50%  P (not white)  0.8 or 80%  A specially designed circuit can only have an output of 110 volts. What is the probability it “feeds” 85 volts?  0 or 0%  Using a regulation deck of cards. What is the probability of choosing a Queen?  (1/13), 0.077, or 7.7%

6.3 What to expect in the long run… THE LAW OF AVERAGES OR LARGE #’s An experiment in which the probability of success in a single trial is p. Suppose that the single trial of this experiment is repeated many times and the outcome of one trial does not affect the outcome of any other trial. The larger the number of trials, the more likely it is that the overall proportion of successes will be close to the probability p. If we roll a number cube six times are we guaranteed to get a one at least one time? Not necessarily. However, based on the law of averages, the more times we roll the number cube the closer the chances of rolling a one will be (1/6) or about 16.67%.

6.3 What to expect in the long run…  EXPECTED VALUE… Is the quantity by which we can perceive each individual result will yield. {Based on the law of averages} MATHEMATICAL EQUATION

6.3 What to expect in the long run…  EXPECTED VALUE…example… Consider selling accident insurance, where if a consumer must quit his/her job due to an injury, they will be paid $100,000. If you can sell 1 million policies at $250 per policy, and according to relative frequency the probability of a policyholder collecting is 1/500. How much profit is expected? Hence, the expected profit is $50 times 1 million or $50 million in profit.

6.3 What to expect in the long run… EEXPECTED VALUE…example2… TAG ALONG LOTTERY EXPECTATIONS Page … If we sum the last column, to find the expected value. This means that you should expect to lose $0.64 per ticket bought. EventValueP(event)Value×P(event) Ticket purchase -$11-$1×1 = -$1 Win free ticket$11/5$1×1/5 = $0.20 Win $5$51/100$5×1/100 = $0.05 Win $1,000$10001/100,000$1000×1/ = $0.01 Win $1 million$1 million1/10,000,000$1million×1/10,000,000 = $0.10

6.3 What to expect in the long run…  GAMBLER’S FALLACY… A concept based on the law of averages that the probabilities of positive events or successes increases after a number of negative events have occurred. An individual has played the slots 299 times, and not won. Is the individual justified to play again given the probability of winning on any particular slot machine is 1 in 300. ANSWER: To base playing again on the previous 299 plays is the gambler’s fallacy.

6.3 What to expect in the long run… HOMEWORK: pg 257 # 1-16 all