The document discusses binomial experiments such as coin flips or free throws, which have yes/no outcomes per trial and a fixed probability of success. It provides the binomial probability formula and explains that

1. Lab 6
Binomial Experiments

2. What are examples of a binomial experiment?
• I shoot 50 free-throws. How many do I make?
• A mother has 7 babies. How many are girls?
• I flip a coin 25 times. How many are heads?
• I call 100 phone #s from the phone book. How many people pick up?
• Each trial results in adding 1 or adding 0 to the total
• The probability of “success” does not change from one trial to the next

3. Guess which ones are binomial experiments
• 150 people play roulette. How many of them win?
• We ask 10 people at random to do 20 pushups. How many of them can do it?
• We ask 10 at random people to do 20 pushups. How many total pushups get
done?
• 30 random people go to Baskin Robbins. How many order vanilla?
• Each trial results in adding 1 or adding 0 to the total
• The probability of “success” does not change from one trial to the next

4. Guess which ones are binomial experiments (part 2)
• 150 people play roulette. How much money do they win?
• 150 people play poker. How many of them leave the table having lost $$?
• We select 10 people at random and measure their height. What is the average
height?
• We select 10 people at random and measure their height. How many people
are over 5’ tall?
• Each trial results in adding 1 or adding 0 to the total
• The probability of “success” does not change from one trial to the next

5. Guess which ones are binomial experiments (part 3)
• 𝑛 students play a game of trying to thread 21 beads onto a stick.
How many of them will succeed in under 1 minute?
• Each trial results in adding 1 or adding 0 to the total
• The probability of “success” does not change from one trial to the next
That’s today’s experiment

6. Why do we like binomial experiments?
Because binomial experiments have a formula for calculating the
probability of getting a certain result:
𝑃 𝑋 = 𝑥 =
𝑛
𝑥
⋅ 𝑝𝑥
⋅ 1 − 𝑝 𝑛−𝑥
where…
𝒙 is the number of “successes” in the experiment
𝒏 is the number of trials in the experiment
𝒑 is the probability of success in each trial

7. What is 𝑛
𝑥
????
𝑃 𝑋 = 𝑥 =
𝑛
𝑥
⋅ 𝑝𝑥
⋅ 1 − 𝑝 𝑛−𝑥
𝑛
𝑥
=
𝑛!
𝑛 − 𝑥 ! ⋅ (𝑥!)
But if you don’t feel like calculating it the long way, there is a
function in your calculator to calculate it

8. What is 𝑛! ????
𝒏! = 𝑛 ⋅ 𝑛 − 1 ⋅ 𝑛 − 2 ⋅ … ⋅ 2 ⋅ 1
𝟓! = 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1 = 120
𝟑! = 3 ⋅ 2 ⋅ 1 = 6
𝟗! = 9 ⋅ 8 ⋅ 7 ⋅ 6 ⋅ 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1 = 362880
For example…
𝑛
𝑥
=
𝑛!
𝑛 − 𝑥 ! ⋅ (𝑥!)

9. Using the binomial formula
For example, if it’s true that a coin is fair, what is the probability
that I get 9 heads out of 10 flips?
where…
𝒙 is the number of “successes” in the experiment = 𝟗
𝒏 is the number of trials in the experiment = 𝟏𝟎
𝒑 is the probability of success in each trial = 𝟎. 𝟓
𝑃 𝑋 = 𝑥 =
𝑛
𝑥
⋅ 𝑝𝑥
⋅ 1 − 𝑝 𝑛−𝑥
𝑃 𝑋 = 9 =
10
9
⋅ 0.59
) ⋅ 1 − 0.5 10−9
= 0.009765625

10. Our binomial experiment
• 𝑛 students play a game of trying to thread 21 beads onto a stick.
How many of them will succeed in under 1 minute?
. RULES .
• Move one bead at a time
• Keep the stick vertical

11. Group Max number of beads moved in
any one attempt
Outcome (success or failure)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16

14. HINT:
On this histogram 
Which values of 𝑥 are most
likely to occur?
𝒙

15. mean 𝑥 = 𝑛 ⋅ 𝑝
HINT: If 𝑥 refers to a Binomial(𝑛, 𝑝) random variable, then
𝑠𝑑 𝑥 = 𝑛 ⋅ 𝑝 ⋅ (1 − 𝑝)
You must replace 𝒏 and 𝒑
with VALUES and calculate.
What are our values for 𝒏
and 𝒑???

16. https://www.slideshare.net/AndersonBussing/lab6-251224855

17. HINT:
If our class value of 𝑥 was 17, we
would write = 17 in those fields
and the bar would highlight red
Draw this histogram (with
highlighted bar) include
numbers and everything

18. 𝑃 𝑋 = 𝑥 =⋅
𝑛
𝑥
𝑝𝑥
⋅ 1 − 𝑝 𝑛−𝑥
You must plug our values for 𝒙, 𝒏, 𝒑 into the below formula.
Show your work.
where…
𝒙 is the number of “successes” in the experiment
𝒏 is the number of trials in the experiment
𝒑 is the probability of success in each trial

19. HINT:
If the 80% was the real proportion of people who can do the bead game
in less than 1 minute, where would we expect our class value for 𝑥 to fall
on the histogram we made?
In the tails of the histogram? In the center?
Now where did our class value actually fall? Did it match what we
expected?

20. HINT:
Based on our where our class’ number of successes 𝑥 fell on the
histogram, do you think the game should be made harder or easier or
kept the same? In order so that 80% of people can complete the task in
time (and thus 20% are unable to complete the task in time).
What can they do to make the game harder/easier?

21. QUESTIONS 8-9:
Raise your hand and ask me if you have doubts about these.