AP Statistics Unit 4 Binomial & Geometric Distributions

Topics 4.10–4.12: Binomial & Geometric Distributions

Recognizing, Calculating, and Interpreting Special Distributions
Video Follow-Along Worksheet
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Learning Objectives:
🖩 Practice on the TI-84 →
🖩 Practice on the TI-84 →bonus

Key Vocabulary

Binomial Setting Binary outcomes, Independent trials, fixed Number of trials, Same probability
Binomial Random Variable X = the number of successes in n independent trials with probability p
Geometric Setting Binary, Independent, Same p, but NO fixed number of trials (count until first success)
Geometric Random Variable X = the number of trials until the first success
Binomial Mean μX = np
Binomial Std Dev σX = √[np(1-p)]
Geometric Mean μX = 1/p
Geometric Std Dev σX = √(1-p)/p

Topic 4.10: Introduction to the Binomial Distribution

[Video: ~10 min]
Hurricane Context: [00:33] Chief Meteorologist Chikage Windler explains that 53% of tropical storms become hurricanes. In a typical season, there are about 12.1 tropical storms, 6.4 hurricanes, and 2.7 major hurricanes.
1. [02:51] A setting involves repeated trials of a random process where four conditions are met. Use the acronym BINS:
Binary: possible outcomes (success or failure)
Independent: trials of the same random process
Number: A number of trials, n
Same probability: Each trial has the probability of success, p
Example: [03:20] The probability a tropical storm becomes a hurricane is 0.53. If the weather service predicts 6 more tropical storms, what is the probability that exactly 5 become hurricanes?
2. [03:38] Check the conditions for this hurricane example:
Two outcomes? (hurricane or )
Independent? (tropical storms don't interact with each other)
Fixed number? , n =
Same probability? , p =
3. [04:20] Define the random variable: X = the number of that develop from 6 tropical storms.
4. [04:30] In a binomial setting, the random variable X (the number of ) is called a random variable.

Binomial Probability Formula [04:59]

P(X = x) = nCx · px · (1-p)n-x
where nCx = "n choose x" = n! / [x!(n-x)!]
5. [05:31] Calculate P(X = 5) for the hurricane example:
P(X = 5) = 6C5 · (0.53)5 · (1 - 0.53)1
P(X = 5) = · (0.53)5 · (0.47)1 =
6. [06:06] Interpret: The probability that exactly of 6 tropical storms will become hurricanes is .
Cumulative Example: [06:15] If 22 tropical storms are predicted, what is the probability that at least 16 become hurricanes?
7. [06:36] A binomial probability involves finding P(X ≥ k) or P(X ≤ k) — summing multiple values.
8. [06:53] For P(X ≥ 16) with n = 22 and p = 0.53:
P(X ≥ 16) = P(X=16) + P(X=17) + ... + P(X=) =
Calculator Note [07:27]:

Warning [08:00]: Writing "binompdf(6, 0.53, 5)" alone on the AP exam will NOT earn credit. You must define the variable and distribution!

9. [09:07] Key Takeaway: When calculating binomial probabilities, you must:
1. Define the
2. Identify the and values of n and p
3. Calculate the correct
4. Answer the asked

Topic 4.11: Parameters for a Binomial Distribution

[Video: ~6 min]
Cell Phone Example: [00:25] Over 5,700 cell phone screens are cracked every hour in the US. According to Digital Trends, 21% of cell phone owners have a cracked screen. A team called "Better Than a Bandage" repairs cracked screens.
10. [00:57] If the team randomly selects 40 cell phone owners, define the random variable:
C = the number of cell phone owners with a .
11. [01:31] Verify binomial conditions:
Binary: Screen is or not cracked
Independent: One person dropping their phone doesn't affect another
n = (fixed)
p = (same for each trial)

Binomial Parameters [01:58]

Mean: μX = np
Standard Deviation: σX = √[np(1-p)]
12. [01:58] Calculate the mean for cracked screens:
μC = np = (40)(0.21) = cell phones
13. [02:25] Interpret the mean: In many random samples of 40 cell phone owners, the team can expect, on , people to have a cracked screen.
14. [02:25] Calculate the standard deviation:
σC = √[np(1-p)] = √[(40)(0.21)(0.79)] = cell phones
15. [02:46] Interpret the standard deviation: In many random samples of 40 cell phone owners, the number with cracked phones will typically from the mean of 8.4 by about phones.
⏸ PAUSE THE VIDEO [03:00]
Try the practice problem before continuing!
Practice Problem: [03:00] The team has supplies for 25 phone repairs. Recent experience shows 15% of people have cracked screens AND will buy a repair. The team plans to approach 150 people. Would you be surprised if they ran out of supplies?
16. [03:40] Define and calculate:
C = number of people who have cracked screens and will buy a repair
n = , p =
μC = (150)(0.15) = people
σC = √[(150)(0.15)(0.85)] = people
17. [04:06] Method 1 — Using standard deviation:
One standard deviation above the mean: 22.5 + 4.37 =
This exceeds the supply of , so running out is surprising.
18. [04:44] Method 2 — Using probability:
P(C > 25) =
This probability is enough that running out is not surprising.

Video 1: The Geometric Distribution (Topic 4.12)

[Video: ~5 min]
Hurricane Example: [00:20] In 2020, NOAA updated their prediction: 41% of tropical storms will become hurricanes. What is the probability that the fourth tropical storm will be the first one to become a hurricane?
19. [00:45] Define the random variable:
H = the number of it takes to get the hurricane.
20. [01:00] H is with p = .
Geometric vs Binomial [01:00]:
Similarities:Binary outcomes, Independent trials, Same probability p
Key Difference:Binomial has n; Geometric has fixed n

Geometric Probability Formula [01:32]

P(X = x) = (1-p)x-1 · p
(x-1) failures, then 1 success
21. [01:32] Calculate P(H = 4):
Storms 1, 2, 3 are failures (not hurricanes): (1 - 0.41)
Storm 4 is success (hurricane): (0.41)
P(H = 4) = (0.59)3(0.41) =
22. [02:12] Cumulative geometric probability:
What is P(first hurricane by the 3rd storm)? P(H ≤ 3) = P(H=1) + P(H=2) + P(H=3)
= (0.41) + (0.59)(0.41) + (0.59)2(0.41) =

Geometric Parameters [03:13]

Mean: μX = 1/p
Standard Deviation: σX = √(1-p) / p
23. [03:13] Calculate the mean for the hurricane example:
μH = 1/p = 1/0.41 = tropical storms
24. [03:13] Interpret: Over many seasons, we can expect it will take about tropical storms, on , to get the first hurricane.
25. [03:40] Calculate and interpret the standard deviation:
σH = √(1-0.41)/0.41 = storms
The number of tropical storms to get the first hurricane will typically differ from the mean of 2.44 by about storms.
⏸ PAUSE THE VIDEO [04:06]
Try the prairie dog practice problem!
Practice: [04:08] 20% of prairie dog litters have 4 or more pups ("large family").
(a) What is P(zoologist waits until 5th litter for a large family)?
(b) How many litters expected until a large family?
26. [04:18] Answers:
(a) P(X = 5) = (0.8)4(0.2) =
(b) μX = 1/0.2 = litters on average

Video 2: Binomial vs Geometric Practice (Topic 4.12)

[Video: ~3 min]
⏸ PAUSE THE VIDEO [00:49]
Work through the Major Hurricanes FRQ-style problem before seeing the solution!
Major Hurricanes Problem: [00:49] Sustained winds of 111+ mph define a major hurricane. There is a 22% chance a tropical storm becomes a major hurricane. NOAA predicts 18 tropical storms this year. A team of pilots is preparing to survey 4 major hurricanes.
27. [00:52] Part (a): What is the probability of NOT having a major hurricane until the 7th tropical storm?
Distribution: with p =
P(X = 7) = (1 - 0.22)6(0.22) = (0.78)6(0.22) =
This probability is , so going this long without a major hurricane would be .
28. [01:26] Part (b-i): Expected number of major hurricanes from 18 storms?
Distribution: with n = and p =
μX = np = (18)(0.22) = major hurricanes
29. [01:45] Part (b-ii): Expected number of tropical storms until first major hurricane?
Distribution: with p = 0.22
μY = 1/p = 1/0.22 = tropical storms
30. [02:01] Part (c): If pilots prepare for 4 major hurricanes, is that sufficient?
P(X > 4) where X is binomial with n = 18, p = 0.22
P(X > 4) = 1 - P(X ≤ 4) =
This probability is , so the pilots may sufficiently prepared.
Key Takeaways [02:34]:

Reflection Question

[AI-Graded]
R1. A meteorologist says there is a 30% chance that a given tropical storm will become a major hurricane (Category 3+). Compare how you would set up a binomial vs. a geometric problem using this context: For each, state the distribution with parameters, set up the calculation (you don't need to compute the final answer), and explain how you know which distribution to use.

Exit Ticket

The National Hurricane Center reports that about 40% of Atlantic hurricanes make landfall in the United States.
  1. In a season with 8 Atlantic hurricanes, what is the probability that exactly 3 make landfall? Show your setup and calculation.
  2. What is the expected number of hurricanes that will make landfall in a season with 8 hurricanes? Interpret this value.
  3. What is the probability that the first hurricane to make landfall is the 4th Atlantic hurricane of the season? Show your setup and calculation.
  4. On average, how many Atlantic hurricanes must form before one makes landfall? Interpret this value.

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