Binomial Breakdown Slides Stats Lab
Binomial Breakdown
Mastering criteria, digital calculations, and expected values.
THE BINS TEST
Four requirements for every binomial setting.
B
Binary
Two outcomes (Success/Failure).
I
Independent
Trials don't affect each other.
N
Number
A fixed number of trials.
S
Success
Prob. stays constant (p).
EXPECTED VALUE
On average, how many successes should we see?
E(X) = n • p
n
Trials
The total number of chances.
p
Probability
The chance of success on one trial.
QUICK CHECK
A manufacturing plant finds that 2% of widgets are defective.
"In a batch of 5,000 widgets, how many defective ones do we expect to see on average?"
5,000 • 0.02 = 100
THE PROBABILITY ENGINE
The mathematics for specific outcomes.
P(X = k) =
nk
• pk • (1 - p)n - k
nCk
Ways
Combinations of successes.
pk
Successes
Probability of k successes.
(1-p)n-k
Failures
Probability of the remaining failures.
FORMULA DRILL
In the exact hand-formula, what does the exponent n - k represent?
The total number of failures.
CLASS DRILL
"Fill in the expression for exactly 3 heads in 5 flips:"
(
C
) • (
)
• (
)
CALCULATION STATION
EXACT MATCH
Exactly 6 makes out of 8 shots (p=0.75)
BINOM.DIST(6, 8, 0.75, FALSE)
MORE THAN
More than 10 pets in 15 houses (p=0.60)
1 - BINOM.DIST(10, 15, 0.6, TRUE)
PRO CHEAT SHEET
n • p
Expected Value
False
Exactly k
True
At Most k
1 - True
More Than k
Binomial Breakdown Answer Key KEY 01
KEY 01. BINS Criteria
B: Binary (Success/Fail)
N: Fixed Trials (n)
KEY 02
KEY 02. Check
YES.
Replacement ensures constant probability (S) and independence (I). Fixed trials n=10. Binary (Heart/Not).
KEY 03
KEY 03. Parameters
n=50
p=0.04
k=4
KEY 04
KEY 04. Boundary Logic
"Fewer than 5" targets outcomes 0, 1, 2, 3, 4. Modern BINOM.DIST cumulative calculates P(X ≤ k), so we must use k=4.
KEY 05
KEY 05. Exact Match
=BINOM.DIST(6, 8, 0.75, FALSE)
0.3115
KEY 06
KEY 06. Cumulative
=BINOM.DIST(3, 10, 0.25, TRUE)
0.7759
KEY 07
KEY 07. Greater Than
=1 - BINOM.DIST(10, 15, 0.6, TRUE)
0.2173
KEY 08
KEY 08. Violations
FAIL: INDEPENDENT (I)
No replacement from a small population (30) changes probability significantly for each subsequent pick.
KEY 09
KEY 09. Overbooked
=BINOM.DIST(20, 20, 0.95, FALSE)
0.3585
KEY 10
KEY 10. Range Logic
P(X ≤ 3) - P(X ≤ 0)
=BINOM.DIST(3, 100, .02, T) - BINOM.DIST(0, 100, .02, T)
0.7225
KEY 11
KEY 11. At Least k
=1 - BINOM.DIST(14, 20, 0.8, TRUE)
0.8042
KEY 12
KEY 12. SCRAP Case
Success = SCRAP (p=0.10)
=1 - BINOM.DIST(2, 25, 0.1, TRUE)
0.4629
KEY 13
KEY 13. Digital Mktg
=BINOM.DIST(30, 200, 0.15, FALSE)
0.0772
KEY 14
KEY 14. Polling
=BINOM.DIST(44, 100, 0.48, TRUE)
0.2403
KEY 15
KEY 15. Trends Range
P(X ≤ 25) - P(X ≤ 19)
=BINOM.DIST(25, 30, .7, T) - BINOM.DIST(19, 30, .7, T)
0.6865
KEY 16
KEY 16. Theory Decode
Represents probability of Failures occurring. (1-p) is the chance of one fail; (n-k) is the number of trials that must be failures.
KEY 17
KEY 17. EV Theory
n • p
Calculates the long-run average (mean) of the distribution.
KEY 18
Binomial Case Cards Binomial Case Cards
Task Cards & Scenario Bank
Activity Resource
Task Card Instructions: Cut along the dashed lines. For each scenario, students must identify n , p , k , the Expected Value E[X] , and write the Google Sheets Formula .
CARD 01
Shopping Spree
A retail store knows that 25% of visitors make a purchase. If 80 people visit the store today, what is the probability that exactly 20 make a purchase?
n: ____
p: ____
k: ____
E[X]: ____
Formula:
CARD 02
Safety Check
A car safety feature has a failure rate of 0.1%. In a shipment of 5,000 cars, what is the probability that at least 1 car has a faulty feature?
n: ____
p: ____
k: ____
E[X]: ____
Formula:
CARD 03
App Usage
65% of students use a study app. If you interview 40 students, what is the probability that between 25 and 30 (inclusive) use the app?
n: ____
p: ____
k: ____
E[X]: ____
Formula:
CARD 04
Cafe Orders
A cafe finds that 40% of customers order a pastry. In a group of 12 customers, what is the probability that more than 5 order a pastry?
n: ____
p: ____
k: ____
E[X]: ____
Formula:
CARD 05
Shipping
98% of packages arrive on time. Out of 150 packages, what is the probability that at most 145 arrive on time?
n: ____
p: ____
k: ____
E[X]: ____
Formula:
CARD 06
Germination
A seed packet has a 90% germination rate. If you plant 100 seeds, what is the probability that fewer than 85 sprout?
n: ____
p: ____
k: ____
E[X]: ____
Formula:
CARD 07
Ad Recall
30% recall an ad. If 200 watch, what is the probability that exactly 65 recall it?
n: ____
p: ____
Binomial Breakdown Worksheet slice edition CARD 01
Binomial Breakdown
Part 1: BINS Criteria
1. Define the four requirements for a binomial setting:
B:
I:
N:
S:
CARD 02
2. Scenario Check
"A student picks a card from a deck, records the suit, and replaces it. They repeat this 10 times. Find the probability of exactly 3 Hearts."
Does this meet BINS? Explain:
CARD 03
3. Parameter Identification
"A quality engineer inspects 50 microchips. 4% are defective. Find the probability that fewer than 5 are defective."
n
p
k (Boundary)
CARD 04
4. Boundary Logic
In Card 03, why did you choose your specific value for k ? Explain the logic for finding "fewer than 5" in a discrete setting.
CARD 05
5. Exact Match
Basketball player: 75% accuracy. 8 shots. Probability of exactly 6 ?
Modern Spreadsheet Formula
=BINOM.DIST(
P = ______
CARD 06
6. Cumulative "At Most"
Guessing on 10-question quiz (4 options). Probability of at most 3 correct?
Modern Spreadsheet Formula
=BINOM.DIST(
P = ______
CARD 07
7. Greater Than
60% own pets. Sample of 15. Probability that more than 10 own a pet?
k logic: k = ____
=1 - BINOM.DIST(
P = ______
CARD 08
8. Setting Violations
Selecting 5 students from a class of 30 without replacement . Why does this fail BINS?
CARD 09
9. Overbooked Flight
"20 tickets sold. 19 seats. 95% show up. Probability the flight is overbooked?"
k logic: k = ____
=BINOM.DIST(
P = ______
CARD 10
10. Range Logic
100 lightbulbs. 2% failure rate. Find the probability that between 1 and 3 (inclusive) are defective.
Step 1: Range Logic (Symbolic)
P(X ≤ ) - P(X ≤ )
Modern Spreadsheet Formula
=
CARD 11
11. At Least k
Medical trial success = 80%. 20 patients. Probability that
Binomial Blitz Quiz Binomial Blitz Quiz
Assessment: Binomial Distribution & Modeling
Student:
Date:
01
Which requirement of the BINS criteria is violated if you are selecting 5 marbles from a bag of 20 without replacement?
Binary (B)
Independent (I)
Number of trials (N)
Success Probability (S)
02
A digital ad has a click-through rate of 5%. If the ad is shown to 1,200 people, how many clicks are expected on average?
Show Work & Answer:
03
Which Google Sheets formula correctly calculates the probability of getting exactly 8 successes in 20 trials with p = 0.45?
=BINOM.DIST(8, 20, 0.45, TRUE)
=BINOM.DIST(20, 8, 0.45, FALSE)
=BINOM.DIST(8, 20, 0.45, FALSE)
04
To find the probability of getting at least 5 successes, which symbolic logic is required?
1 − P(X ≤ 5)
1 − P(X ≤ 4)
P(X ≤ 5) − P(X ≤ 4)
05
In the probability formula P(X=k) = (nCk) • pk • (1−p)n−k, what does the term (1−p) represent?
06
You want the probability that between 12 and 15 people (inclusive) own a pet in a sample of 50. Write the symbolic logic using P(X ≤ k) notation:
P(X ≤
) − P(X ≤
)
07
"A researcher polls 500 citizens to see if they approve of a new park." Identify the value of the parameter n :
08
Set up the hand-calculation expression for the probability of 2 successes in 5 trials where p = 0.2. (Do not solve):
P(X = 2) =
5 2
• ( 0.2 )
• ( 0.8 )
09
Which parameter represents the boundary value (the specific number of successes) you enter into a spreadsheet formula?
Number of trials (n)
Success Probability (p)
Number of successes (k)
10
Explain why an expected value of 7.2 does not mean that 7.2 people will succeed in any single sample.
Standard: S-CP.B.9 • S-MD.A.2
© 2026 Lenny Educational Designs
Binomial Blitz Quiz Answer Key Answer Key
Binomial Blitz Quiz
Teacher Resource
01
Requirement Violated
Independent (I)
Explanation: Selection without replacement means trials are dependent (probabilities change), violating the Independence requirement for a binomial setting.
02
Calculation
60 clicks
Logic: E(X) = n • p = 1,200 • 0.05 = 60
03
Correct Syntax
=BINOM.DIST(8, 20, 0.45, FALSE)
Note: FALSE is required for a point probability (exactly k).
04
Boundary Logic
1 − P(X ≤ 4)
Explanation: "At least 5" includes {5, 6, 7...}. The complement is {0, 1, 2, 3, 4}.
05
Concept
Probability of Failure (q)
Calculated as (1 − p).
06
Range Logic
P(X ≤ 15) − P(X ≤ 11)
Note: Subtracting up to 11 preserves the "12" in the 12-15 range.
07
Parameter Identification
n = 500
08
Expression Setup
(5C2) •
(0.2) 2
•
(0.8) 3
09
Parameter
k = number of successes
Also known as the boundary or specific outcome being tested.
10
Conceptual Mastery
"Since a binomial variable is discrete, only whole-number outcomes (e.g., 7 or 8) are possible in a single trial. The expected value (7.2) is a theoretical mean—the average result expected over a long sequence of many repeated samples."
