Stat 20: Intro to Probability and Statistics

Transcription

1 Stat 20: Intro to Probability and Statistics Lecture 17: Using the Normal Curve with Box Models Tessa L. Childers-Day UC Berkeley 23 July 2014

2 By the end of this lecture... You will be able to: Draw and interpret a probability histogram for the sum draws from a box Predict what a probability histogram for the sum of draws from a box will look like Use the Central Limit Theorem to calculate probabilities without having to draw a histogram 2 / 19

3 Recap: Box Models & The Normal Curve Box models are useful in analyzing games of chance Draw a box model to describe a process (sums or classify/count) Calculate EV and SE Use normal curve to calculate probability that actually observe some event But why use normal curve? 3 / 19

4 If you have enough draws/trials... The chances become normal. We ll see this via experimentation. Say we are flipping a fair coin. What does our box look like? What if we want to count the number of heads? What is P(Heads)? P(Tails)? What about if there are 2 tosses? What about more tosses? 4 / 19

5 Summarize a probability table A histogram will summarize the probabilities well. What should go on the x-axis? What should our classes be? What about the y-axis? 5 / 19

6 Summarize a probability table (cont.) What did our experiment show us? As the number of draws, the becomes more like a normal curve Probability histograms represent Height represents percent per Make probability histogram more normal by 6 / 19

7 What if the box changes? With the coin flips, the probability histogram for the number of heads began to look normal fairly quickly. What about other scenarios? Add another face, $1 for head, $0 for tail, $15 for new face Sum of dice Number of 6 s Net gain on bet that pays 2 to 1. Win if a single draw from a deck is black. Net gain on bet that pays 25 to 1. Win if a single draw from a deck is a Queen. Others? 7 / 19

8 What have we learned? Does this work for any function of the draws? Does this work for any box? Does this work for any number of draws? How many draws do we need? 8 / 19

9 Probability Histogram Normal Curve With enough draws, the probability histogram for the sum of draws looks normal. Why is this useful? To use the normal curve we need: z = value center spread Value Center Spread 9 / 19

10 The Value(s) Draw a picture, shade an area What do we want to draw? What do we want to shade? Why do we shade this area? When do we use this technique? 10 / 19

11 The Center Draw a picture, shade an area. Where is the center of our picture? Is our picture symmetric? How big is a typical draw? How big is a typical sum? 11 / 19

12 The Spread Draw a picture, shade an area. How spread out is our picture? How far from the typical draw is an individual draw going to be (on average)? How far from the typical sum is an individual sum going to be (on average)? 12 / 19

13 Put it all together Want the chance a sum is between val 1 and val 2 1 Draw a picture, shade area between val and val Convert to z-scores: z 1 = (val 1.5) EV for sum SE for sum and z 2 = (val 2 +.5) EV for sum SE for sum 3 Use normal table to find area 13 / 19

14 The CLT Just a fancy way of saying what we already know. When drawing a random, with replacement, from a box, if there are enough draws, the probability histogram looks normal. The associated normal curve: is centered at the EV for the sum has spread equal to the SE for the sum 14 / 19

15 Example: Playing the Slots A slot machine contains 3 drums that turn rapidly, and then stop on a random character. Each drum has 3 characters: a bar, a cherry, and a diamond. The first drum contains 5 characters (a bar, 2 cherries, and a diamond). The second drum contains 6 characters (2 bars, 3 cherries, and a diamond). The third drum contains 7 characters (3 bars, 4 cherries, and a diamond). What is the chance that in one spin you see all 3 diamonds? What is the chance that in one spin you see all 3 characters the same? 15 / 19

16 Example: Playing the Slots (cont.) A slot machine contains 3 drums that turn rapidly, and then stop on a random character. First drum: a bar, 2 cherries, and a diamond. Second drum: 2 bars, 3 cherries, and a diamond. Third drum: 3 bars, 4 cherries, and a diamond. Suppose it costs $5 to play the slot machine. The machine pays $100 if you see all 3 diamonds. It pays $10 if you see either all 3 bars or all 3 cherries. Make a box model for your net gain when playing this machine. Make a box model for counting the number of times you win when playing this machine. 16 / 19

17 Example: Playing the Slots (cont.) A slot machine contains 3 drums that turn rapidly, and then stop on a random character. First drum: a bar, 2 cherries, and a diamond. Second drum: 2 bars, 3 cherries, and a diamond. Third drum: 3 bars, 4 cherries, and a diamond. You decide to play this game 10 times. Relate your net gain for 10 plays to a box model. About how much do you expect to win? About how much is this likely to be off by? Calculate the chance that you lose between $25 and $50 playing 10 times. Do you think your calculation is accurate? 17 / 19

18 Example: Playing the Slots (cont.) A slot machine contains 3 drums that turn rapidly, and then stop on a random character. First drum: a bar, 2 cherries, and a diamond. Second drum: 2 bars, 3 cherries, and a diamond. Third drum: 3 bars, 4 cherries, and a diamond. You and a group of 9 friends decide to play on similar machines. You each play 100 times. Relate your net gain (as a group) to a box model. About how much do you expect to win (as a group)? About how much is this likely to be off by? Calculate the chance that you lose between $2500 and $3000 playing as a group. Do you think your calculation is accurate? 18 / 19

19 Important Takeaways A probability histogram displays probabilities, not data With enough draws, a probability histogram for the sum of draws with replacement looks like a normal curve How big is enough? Influenced by the make-up of the box The normal curve is centered at the EV for the sum, with spread equal to the SE for the sum We can use the continuity correction, combined with the normal curve, to find probabilities under the normal curve (approximate probabilities from the probability histogram) Next time: What if we re not gambling, but sampling? 19 / 19