Probability
Fill in the blank
Two events are ___ if they have no
outcomes in common.
Disjoint or mutually exclusive
Family with two children, equally likely boy/girl. Probability both are boys given at least one is a boy
BB, BG, GB, GG.
Given at least one boy excludes GG →
three equally likely: BB, BG, GB.
Probability = 1/3
What is uniform distribution?
Each of the values tends to occur with the same
frequency
The histogram looks flat
A college wants to create 4-digit student locker codes. Each digit can be any number from 0 to 9, and digits can repeat.
10^4=10,000
Two events E and F are __ if
the occurrence of event E in a
probability experiment does not affect
the probability of event F.
Independent Events
At a college:
60% of students study late at night
Among late-night students, 70% drink energy drinks.
Among morning students, 30% drink energy drinks.
Question:
If you randomly meet a student who is drinking an energy drink, what is the probability that they study late at night?
P(Late Night | Energy Drink)=.7*0.6/.54=0.42/.54≈0.7778
Label a boxplot.
Q1
Median
Q3
Whiskers
Outliers
LF Min Q1 Median Q3 Max UF
l---------|-----|--------|---------|-----| ---------l
Number of pills in a bottle, what kind of variable is that?
A Discrete Variable, as it is a countable number.
ex. 1,2,3,4,5,6,7
A student council has 7 members, and they need to choose a President, Vice President, and Secretary.
P(7,3) = 210
A variable has a skewed left distribution when?
The tail to the left is longer than the tail to the right
The arrow from the middle to the long tail points left
Arrangements of (Real or Imaginary) can be formed from the letters in the word TENNESSEE.
9! / (1!4!2!2!) = 3,780
T → 1
E → 4
N → 2
S → 2
What are the Measures of Central Tendency?
Measures of Central Tendency are statistical values that describe the center or “typical” value of a dataset. They give you an idea of where the data is concentrated.
Mean, Median, Mode, Distribution shape.
Can researchers claim causation in an observational study?
No
Causation means that one variable directly causes a change in another variable, making it controlled.
Probability 3 drawn balls are same color (6 red, 4 blue, 5 green; draw 3)
(6 CR 3) = 20
(4 CR 3) = 4
(5 CR 3) = 10
TOTAL = 34.
(15 CR 3) 455.
34/455 = 7.47%
60th percentile of
1, 3, 4, 7, 8, 15, 16, 19, 23, 24, 27, 31, 33, 34
(60/100) * (14-1) = 9
9th value= 23
At a college:
55% of students are coffee lovers, and 45% are tea lovers.
40% of coffee lovers buy extra-large drinks.
25% of tea lovers buy extra-large drinks.
Question:
What percent of students, overall, buy extra-large drinks?
Use the Total Probability Rule:
=(0.40)(0.55)+(0.25)(0.45) = 0.22+0.1125 =0.3325 / 33.35%
A group of 12 friends went out for ice cream, and each ordered a different number of sprinkles on their cone. Here’s what they got:
0,5,10,5,20,15,5,0,10,25,15,50, 5, 10, 5, 20, 15, 5, 0, 10, 25, 15, 50,5,10,5,20,15,5,0,10,25,15,5
Organize the data into a frequency table.
Find the relative frequency.
Identify the most popular number of sprinkles.
Sprinkles
Sprinkles Frequency Relative Frequency
0 2 2/12 ≈ 0.167
5 4 4/12 = 0.333
10 2 2/12 ≈ 0.167
15 2 2/12 ≈ 0.167
20 1 1/12 ≈ 0.083
25 1 1/12 ≈ 0.083
What is a stratified sample??
A stratified sample is obtained when we choose
a simple random sample from subgroups of a
population
This is appropriate when the population is made up of
nonoverlapping (distinct) groups called strata
Within each strata, the individuals are likely to have a
common attribute
Between the strata, the individuals are likely to have different attributes
A college is creating a photo lineup for 6 distinct student council members: Alice, Bob, Carol, Dave, Eve, and Frank.
Rules:
Only 4 students will be in the photo.
Alice must be in the first position.
First position is already taken by Alice, so we now have 3 positions left to fill.
5!/(5−3)!=120/2=60
Which sampling method ensures every subgroup of the population is represented?
Random Sampling
A committee of 5 chosen from 8 men and 7 women. The Probability Committee has more women than men.
COMBINATION
Total ways = (15 CR 5)=3003.
3 women: (7 CR 3)(8 CR 2)=35⋅28=980.
4 women: (7 CR 4)(8 CR 1)=35⋅8=280.
5 women: (7 CR 5)(8 CR 0)=21⋅1=21.
Sum = 980 + 280 + 21 = 1281.
Probability 1281/3003=427/1001
The following are the test scores of 12 students on a math exam:
55,60,62,65,68,70,72,75,78,80,85,90
Identify the minimum, Q1, Q2, Q3, IQR and maximum.
Are there any outliers?
Q1=26/2+65=63.5
Q3=278+80=79
Median/Q2 =270+72=71
QR = Q3 − Q1 = 79 − 63.5 = 15.5
Lower bound = Q1 − 1.5 × IQR = 63.5 − 23.25 ≈ 40.25
Upper bound = Q3 + 1.5 × IQR = 79 + 23.25 ≈ 102.25
NO OUTLIERS
Minimum = 55
Maximum = 90
How do you make a STEM-AND-LEAF PLOT?
Draw an example or describe.
STEM-AND-LEAF PLOT
To draw a stem-and-leaf plot, each data value
must be broken up into two components
The stem consists of all the digits except for the right
most one
The leaf consists of the right most digit
For the number 173, for example, the stem would be
“17” and the leaf would be “3”
A pizza club at college has 8 different toppings.
The club wants to make a special pizza with 3 toppings in a specific order.
The club also wants to make a pizza with 3 toppings where order doesn’t matter.
Questions:
a) How many ways can they choose the 3 toppings in order?
b) How many ways can they choose the 3 toppings without caring about order?
c) Are the answers are different?
Part a) Ordered toppings → Permutation
Order matters →P(8,3)=(8−3)!8!=8!/5!=336
Part b) Unordered toppings → Combination
Order doesn’t matter → 56C(8,3)=8!/5!3!=56
Part c) Explanation
Permutation counts every arrangement (e.g., pepperoni-cheese-mushroom is different from mushroom-cheese-pepperoni).
Combination counts only the set of toppings, ignoring order.
A college is hosting a pizza party for 10 students.
The students are lining up to get pizza.
The first 3 in line get free soda.4 of the 10 students are vegetarians.
The order of students in line matters
What is the probability that all 3 students who get free soda are vegetarians?
Choose 3 vegetarians from 4 P (4,3)=4⋅3⋅2=24
Arrange the remaining 7 students → 7!=50407! = 50407!=5040
10 students = 10! = 3,628,800
Favorable arrangements= 24 ⋅ 5040=120,960
120,960/ 3,628,800≈0.0333