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Chapter 17: The binomial model of probability Part 2

Chapter 17: The binomial model of probability Part 2. AP Statistics. The binomial model: jumping in with both feet. There’s a close connection between tree diagrams, combinations, expanding binomial expressions, and the binomial model

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Chapter 17: The binomial model of probability Part 2

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  1. Chapter 17:The binomial model of probabilityPart 2 AP Statistics

  2. The binomial model:jumping in with both feet • There’s a close connection between tree diagrams, combinations, expanding binomial expressions, and the binomial model • I’ve debated about giving you a lot of background on it to show you the connections FIRST • Instead, I think it will be more effective to show you how to solve the problems, and then show you the connections, which then may be more meaningful.

  3. The binomial model:formulae • So here they are, the three formulae you need to recognize and use. • Let’s do the easiest two first, mean and standard deviation • Check out the Math Box on p. 392 if you want to see the derivations • Mean: • Standard deviation:

  4. The binomial model:probability of exactly k successes in n trials • This is the key equation to recognize (memorizing it is hard, at least at first) • Remember that p=probability of success, q=probability of failure, and the binomial coefficient is the number of combinations that satisfy k successes in n trials • Here it is……wait for it…….

  5. The binomial model:how to dissect the binomial formula • It’s the number of combinations you can get from taking k samples from a population of n • That, in turn, is equal to • These are the calculations we’ll be working on today

  6. The binomial model/Exercise 13(d):Starting with Exercise 13(d) • We were not able to do Exercises 13(d)-(f) or 14(d)-(f) because we needed to apply the binomial model. • So let’s start with Exercise 13(d), which asks for what percentage of the populations has exactly 3 lefties. • Before going on to the next slide, work for 2-3 minutes and see how many different combinations you can make using three lefties and two righties.

  7. The binomial model/Exercise 13(d):combinations of 3 lefties, 2 righties • Here’s the distribution on the right. • It took me about 5 minutes to derive and check it; YMMV. • But even if you do it faster than I did, what about 3 out 10? Or 3 out of 15? Or even 3 out of 8? • It will take way too long. • We need a better (faster) method. • L LL R R • L L R L R • L L R R L • L R L L R • L R L R L • L R R L L • R L LL R • R L R L L • R L L R L • R R L LL

  8. The binomial model/Exercise 13(d):Using the binomial coefficient • The binomial coefficient will save us from the tedium of finding such distributions • For n=5 and k = 3 (the conditions of our problem), we can calculate the number of combinations directly:

  9. The binomial model/Exercise 13(d):Applying the binomial coefficient to the probabilities • We have the binomial coefficient, which is 10 for this problem. • Now, how to apply the q and p probabilities? • The best thing to do is to create a table. We’re going to spend some time doing this, and it should help to make everything clear.

  10. The binomial model/Exercise 13(d):Making a probability table • Because the calculations are complicated, we’re going to make a couple of tables to help make things clear. • Here’s the form of the tables to copy for this part of the exercise. Make two of them:

  11. The binomial model/Exercise 13(d):Filling in the 1st probability table with formulae • Note that the table k=0 through k=5. We’ll discuss why in a bit as we fill it in. • Now, let’s fill in the formulas before you do the calculations. You’ll do the calculations in the 2nd table. Fill in the blanks as indicated:

  12. The binomial model/Exercise 13(d):What the 1st table should look like when you’re done

  13. The binomial model/Exercise 13(d):Starting the 2nd table • Your second table will contain the calculations using the formulas in the 1st table. As you start, it should look like this:

  14. The binomial model/Exercise 13(d):Completing the 2nd table • Using the formulas in Table 1, do the calculations. Be sure to fill in the calculations in the second column from the right that I’ve grayed out. • By the end, you should have a table of all the values possible where there are 0 lefties, 1 lefty, …., 4 lefties, and 5 lefties. • It will probably take about 10 minutes to complete this. Be sure to complete the totals for the columns headed by (5 n) and by (5 k)q5-npn.

  15. The binomial model/Exercise 13(d):What the second table should like and its meaning

  16. The binomial model/Exercise 13(d):Important points • (5 k)q5-kpk is the probability that you will get k events in 5 trials. • The column headed by (5 k)q5-kpkadds up to 1, which tells us that this is the TOTAL probability model. • Note that we start with k=0….we DO have the probability that we will get zero lefties. • If you have k trials, you will have k+1 entries in your table.

  17. The binomial model/Exercise 13 (e):Using the table to determine summed probabilities • 13(e): at least 3 lefties in the group. • That means there could be 3….or 4…..or 5. • Find the total by adding the areas marked:

  18. The binomial model/Exercise 13 (f):Using the table to determine summed probabilities (2) • 13(d): no more than 3 lefties in the group. • That means there could be 3, 2, 1, or 0 • Find the total by adding the areas marked:

  19. The binomial model:Answers to Exercises 13(e) and 13(f) • Exercise 13(e): 0.982 • Does this answer make sense? • Exercise 13(f): 0.0179 • Does THIS answer make sense?

  20. The binomial model/Exercise 14:Preparing the chart • Same as 13, except we now have 6 arrows, not 5 people • Do the tables again, starting with a blank table like this: Go to the next slide when done.

  21. The binomial model/Exercise 14:The complete chart • Here’s what you should have come up with. • The answers are similar to what you did for 13. Use your chart to calculate them. Answers on next slide.

  22. The binomial model/Exercise 14:Answers • 0.246 (yellow shading) • 0.0179 (adding numbers circled in red • 0.345 (adding numbers circled in green)

  23. The binomial model:Review of how to calculate combinations • Remember the formula: • k is the number of events you’re looking for in the population n • Be sure you put the k with the p, which, as you’ve seen, may change from problem to problem. • Don’t forget to calculate all three parts of the equation and do the multiplication

  24. The binomial model:Practice: Exercise 19 • Similar to what we’ve done already • Tennis player has successful first serve 70% of the time. • Assuming independence and 6 serves (n=6), calculate the following probabilities: • All 6 serves go in • Exactly 4 serves go in • At least 4 serves go in • No more than 4 serves go in • Spend 5-10 minutes calculating (a) through (c), then advance to the next slide for the analysis.

  25. The binomial model:Practice: Exercise 19(a) • All six serves go in. • n=6, k=number of good serves=6, p=0.7, so q=0.3 • Calculate combinations first: • Next, pkqn-k=(0.7)6(0.3)6-6=(0.7)6(0.3)0=(0.7)6=0.118

  26. The binomial model:Exercise 19(b) • Exactly 4 serves go in. • Similar to (a), but with different numbers: n still 6, but k=4 • Binomial coefficient: • Next, pkqn-k=(0.7)4(0.3)6-4 = (0.7)4(0.3)2 =(0.24)(0.09)=0.0216 • Multiply 15 times 0.0216 to get 0.324

  27. The binomial model:Exercise 19(c) • At least 4 serves in means she got 4, 5 or 6 serves in. If we know probabilities, we can add them. • Already know 6 from (a) and 4 from (b) • Calculate probabilities of 5: • Calculate pkqn-k (continued on next slide)

  28. The binomial model:Putting all of 19(c) together • Remember that the total probability of at least 4 serves going in is P(4)+P(5)+P(6) • From previous slides, we have • P(4)= 0.324 • P(5)=0.301=6×0.0504 (from slide 27) • P(6)=0.118 • TOT: 0.743 • Whenever you have multiple outcomes, calculate and add them together!!

  29. The binomial model:Homework • Chapter 17, Exercises 15, 16, 17 18 and 20.

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