Chapter 7 Day 3

Warm - Up Jill sells charm bracelets. The table below shows the distribution of X the number of charms sold per bracelet. Jill sells charm bracelets. The table below shows the distribution of X the number of charms sold per bracelet. Does this represent a legitimate distribution? Does this represent a legitimate distribution? Find the mean μ X for the number of charms sold per bracelet. Find the mean μ X for the number of charms sold per bracelet. Find the standard deviation σ X Find the standard deviation σ X Number of Charms Probability

Homework Solutions 8. μ = 2.25; σ = A. will go over! B. μ = 0.75; σ = C. Casino keeps $.25 of every $1 C. Casino keeps $.25 of every $1 10. A. μ = 2.68; σ = B. will share examples B. will share examples

Question 11 Let’s do the simulation together! Let’s do the simulation together! What is the mean outcome? What is the mean outcome? Is it to your advantage to play? Is it to your advantage to play?

Recall the Law of Large Numbers Draw independent observations at random from any population with finite μ. Decide how accurately you would like to estimate μ. As the number of observations increases, the mean of the observed values eventually approaches the mean of the population as closely as you specified and then stays that close. Draw independent observations at random from any population with finite μ. Decide how accurately you would like to estimate μ. As the number of observations increases, the mean of the observed values eventually approaches the mean of the population as closely as you specified and then stays that close.

Recall the Law of Large Numbers

The “law of small numbers” The rules of probability and the law of large numbers describe the regular behavior of chance phenomenon in the long run. The rules of probability and the law of large numbers describe the regular behavior of chance phenomenon in the long run. We tend to expect even short sequences of random events to show the kind of average behavior that only appears in the long run. We tend to expect even short sequences of random events to show the kind of average behavior that only appears in the long run.

The “law of small numbers” If a basketball player makes several consecutive shots, both the fans and his teammates believe that he has a “hot hand” and is more likely to make the next shot. If a basketball player makes several consecutive shots, both the fans and his teammates believe that he has a “hot hand” and is more likely to make the next shot. In reality, each shot is independent. Players typically perform consistently, not in streaks. In reality, each shot is independent. Players typically perform consistently, not in streaks. Gamblers also follow the “hot hand” theory and continue betting – this leads to lots of money for the casinos Gamblers also follow the “hot hand” theory and continue betting – this leads to lots of money for the casinos If you flip a coin 10 times you may get a streak of 7 heads, but keep flipping 10,000 more times and there will be thousands of heads and tails to balance it out. If you flip a coin 10 times you may get a streak of 7 heads, but keep flipping 10,000 more times and there will be thousands of heads and tails to balance it out.

Classwork Work on Questions Work on Questions 12-14

Answers 12. Trials are independent so a streak doesn’t mean that red is necessarily “hot”. With the cards, he is incorrect because the probability of red and black depends on the cards that have already been dealt. 12. Trials are independent so a streak doesn’t mean that red is necessarily “hot”. With the cards, he is incorrect because the probability of red and black depends on the cards that have already been dealt. 13. no, “at bats” are independent and the 35% figures for the long run, not the short run. 13. no, “at bats” are independent and the 35% figures for the long run, not the short run. 14. A. independentB. not independent 14. A. independentB. not independent C. not independentD. not independent C. not independentD. not independent E. independent E. independent