Binomial distribution

Learning objectives (and summaries)

Recognize the binomial setting and calculate probabilities of binomial events.

- Understand that the binomial setting is the basis for the normal curve
- As n->infinity and p=0.5, the distribution becomes entirely normal. A large number of 50/50 choices that lead to an overall outcome is why the normal distribution is so common.
- Recognize the binomial setting from a non-binomial setting
- An event with a fixed number of trials and a fixed probability of success for each trial.
- Read P(X>4) notation for the probabilities of discrete events.
- P(X=4): the probability that you get exactly 4 successes
- P(X≥4): the probability that you get 4 or more successes
- P(X>4): the probability that you get more than 4 successes (5 or more)
- P(X≤4): the probability that you get 4 or fewer successes
- P(X<4): the probability that you get less than 4 successes (3 or fewer)
- Find the probability of a binomial event using binomcdf(n,p,x) or binompdf(n,p,x) on a TI-83.
- Exactly one value: binompdf(n=number of trials,p=probability of success,x=how many successes)
- Cumulative range of values: binomcdf(n=number of trials,p=probability of success,x=how many successes OR FEWER)

Assessment

- Test (8 elective pts): 5 questions (2 short answer/MC, 3 numeric); 1 of these AP free response questions (3pts):
  - See AP Formula Sheet Here: tinyurl.com/apstatssheet
- AP FRQ 2004 #3 (solution)
  - AP FRQ 2010B #3 (solution)
- Free throws as binomial trials project (5 elective pts)
  - Complete worksheet at bottom of page (print and write or use a PDF app)
- Plinko (up to 8 elective pts)
  - Create a short video that shows how the binomial distribution can predict the probable outcomes in the game Plinko. Explain why the game fits the binomial conditions and perform the calculations to predict how likely the disk is to fall in each slot. Points awarded depend on the quality of the explanation, accuracy of the math, and quality of the video.

Instruction

Practice

Determine if each of the following situations, determine if the count has a binomial distribution. If it does, explain why each condition is met. If it does not, say what condition was not met.

- 1. I flip a coin and count how many times it takes to land a “head”.
- 2. I draw 10 cards from a deck, one at a time, without replacement, and see how many 7’s come up.
- 3. I roll a die 100 times and count how many of each number I get.
- 4. I spin a spinner 12 times and count how many times it lands on the first region.
- 5. I throw a bean bag at a target, receive coaching, and repeat 5 times. I count how many bags hit the target.
- 6. I use a random number generator to determine success or failure. I count how many times until I succeed.

In the following situations, it is common to use the binomial distribution as a model. What cautions should we consider when using it in these situations?

- 7. A basketball player shoots 25 free throws and counts her made shots.
- 8. A baseball player throws 80 pitches and counts his strikes (two reasons).

Plug and chug: use your calculator to find the probability of a single value in the following scenarios.

- 9. 5 trials, p=0.3, find P(X=4).
- 10. 8 trials, p=0.625, find P(X=2).
- 11. 15 trials, p=0.5, find P(X=6).
- 12. 100 trials, p=0.1, find P(X=10).

First, underline the values you are looking for in the scenario. Then, write a statement of how you would use your calculator to find the probability of the range of values. Finally, calculate.

- 13. 8 trials, p=0.625, find P(X≤5).
- 14. 8 trials, p=0.625, find P(X>5).
- 15. 8 trials, p=0.625, find P(X=5).
- 16. 8 trials, p=0.625, find P(X≥5).
- 17. 8 trials, p=0.625, find P(X<5).

Write a calculator statement for each of the following scenarios. Then calculate the probability.

Billy makes 77% of his free throws. His probability of making the shot is fairly constant across different games. If Billy shoots 48 free throws this season…

- 18. How many free throws do you expect Billy to make? Round to the nearest integer.

19. Using your answer for for the expected number (the mean), how likely is Billy to make exactly this number of free throws?
- 20. How likely is he to make at least 34 free throws?
- 21. How likely is he to make more than 41 free throws?
- 22. How likely is he to make at most 34 free throws?
- 23. How likely is he to make fewer than half of his free throws?

Beth wants to do a sampling experiment on a city where 32% of the people prefer to not eat meat. Beth takes a SRS of 120 people.

- 24. What is the probability that she finds less than 34 who don’t prefer meat?
- 25. What is the probability that she finds more than 43 who don’t prefer meat?
- 26. How confident are you (i.e. what is the probability) that Beth will find 34 to 43 people who don’t prefer meat?

EXTRA: First, underline the values you are looking for in the scenario. Then, write a statement of how you would use your calculator to find the probability of the range of values. Finally, calculate.

1. 5 trials, p=0.3, find P(X≤3).
2. 5 trials, p=0.3, find P(X<3).
3. 5 trials, p=0.3, find P(X≥3).
4. 5 trials, p=0.3, find P(X>3).
5. 5 trials, p=0.3, find P(X=3).

EXTRA: Mary is a consistent, expert bean bag tosser who lands 61% of her throws in the hole. If Mary throws 100 bags…

6. What is the mean of this binomial distribution? What does the “mean” mean in the context of this situation?
7. What is the standard deviation of this distribution?
8. What number of made throws is two standard deviations below the mean? Find the probability that Mary makes fewer than this number.
9. What number of made throws is two standard deviations greater than the mean? Find the probability that Mary makes more than this number.
10. How likely is Mary to make exactly 62 throws?
11. How likely is Mary to make at most 50 throws?

EXTRA: Jack wants to do a sampling experiment on a classroom where 57% of the people prefer Batman over Iron Man. Jack takes a random sample of 12 people.

12. What is the probability that he finds 4 or fewer Batman fans?
13. What is the probability that he finds 10 or more Batman fans?
14. How confident are you (i.e. what is the probability) that Jack will find 5 to 9 Batman fans?

EXTRA: Practice crunching binomial probabilities.

15. 9 trials, p=0.5, find P(X≤9).
16. 2 trials, p=0.3, find P(X<1).
17. 19 trials, p=0.25, find P(X≥5).
18. 100 trials, p=0.9, find P(X>90).
19. 51 trials, p=0.44, find P(X=40).
20. 12 trials, p=0.4, find P(X≤5).
21. 18 trials, p=0.8, find P(X>10).
22. 42 trials, p=0.23, find P(X=12).
23. 155 trials, p=0.52, find P(X≥70).
24. 805 trials, p=0.78, find P(X<5).

Practice solutions