Math 140 Notes Chapter 5 Modeling Variation With Probability
http://www.youtube.com/watch?v=2y3PH4SqmlA
http://www.youtube.com/watch?v=mhlc7peGlGg&feature=related The Monty Hall
Problem-You Tube
5.1 Randomness
Random: no predictable pattern occurs, no outcome is more likely to occur
Theoretical Probability: long run probability or long run relative frequency; infinitely many
trials
Empirical Probability: short run probability or short run relative frequency; fixed number
of trials
Example:
What is the probability of getting heads when you flip a coin? (this is the theoretical
probability).
Empirical Probability: Let’s flip a coin 10 times & 20 times.
Simulations: replicating trials using more convenient methods such as technology

Rossman Chance applet for simulating coin toss with 10 & 1000 times. (coin
tossing applet)
5.2 Finding Theoretical Probabilities
P(A): Probability of event A occurring
Ex. P(heads) = _____
0 ≤ P(A) ≤1
or 0% ≤ P(A )≤ 100%
P(A does not occur) = P(𝐴𝐶 ) or P(𝐴̅) = 1 – P(A)
Ex. Roll a die. What is the probability of rolling a 5?
P(5) =
P(5𝐶 ) or P(not 5) =
Sample Space: all possible outcomes
Ex. Die: 1 2 3 4 5 6
Coin: H, T
P(A) =
# outcomes in A
# all possible outcomes
Ex.
P(ace) = ____
Ex. RRRRRGGGWW
P(R)=
P(G)=
P(W)=
P(R) + P(G) + P(W) =
Mutually exclusive: When P(A and B) = 0
Ex. cards P(black card and heart) = ______
And/Or Probabilities
Die: P( even or greater than 4) =
P(even and greater than 4) =
5.3 Associations in Categorical Variables
Conditional Probability: probabilities obtained from a subgroup within the sample
Ex. In the data of COC Math 075 ages, we can find probabilities for women only
𝐴
P(A/B)=probability of event A occurring given that B has occurred = 𝐵
Ex. using table on pg 209: Find the probability that a randomly selected divorced person
had a High School education.
ℎ𝑠
P(HS/Divorced) = 𝑑𝑖𝑣𝑜𝑟𝑐𝑒𝑑 =
Dependent vs Independent:
Ex. Are the events ‘the card is a diamond’ and ‘the card is red’ dependent or independent?
(Is it useful if you know the card is already red?)
P(diamond) = 13/52 = ¼
P(diamond/red) = 13/26 = ½
These are associated, therefore dependent
Ex. Are the events ‘card is an Ace’ and ‘card is a diamond’ dependent or independent?
(Is it useful to know the card is already a diamond?)
P(ace) = 4/52 = 1/13
P(ace/diamond) = 1/13
Given that the card was a diamond did not increase the probability
These events are independent.
To check for independence, compare P(A) with P(A/B) to see if they are the same
(independent) or different (dependent)
Sequences of Independent Events: multiply probabilities
Ex. P(first child boy and second child boy) = .51 x .51 = 0.2601 or 26.01%
Ex. 16 in bk:
67% of teachers are very satisfied with their careers. Three teachers are randomly
selected:
a) What is the prob. that all three are satisfied?
b) Prob. that none are satisfied?
c) Prob. That at least one is satisfied?
a) 0.67 x 0.67 x 0.67 = 0.3008
b) not satisfied: 1-0.67 = 0.33 so 0.33 x 0.33 x 0.33= 0.0359
c) P(at least one)= Prob of one is satisfied or two are satisfied or three are satisfied =
complement of ‘none are satisfied’= 1 – 0.0359 = 0.9641
Ex. 17 p220
If there is a water bottle security catches 95%.
5% of people accidently pack a bottle of water.
Use a tree diagram to find probability that a randomly selected person packs a bottle and
security finds it.
5.4 Empirical Probabilities with Simulations
Using random number generator (on minitab for example p. 239)

Assign 0 and 1 (heads, tails)

Assign evens and odds (heads, tails)
Using applets


http://socr.ucla.edu/ for coin experiment, dice experiment, card experiment
Rossman and Chance applets for coin toss (can be used for two outcomes)
Law of Large Numbers: The empirical probability will get close to the theoretical
probability when an experiment is repeated a large number of times.
Note: outcomes are never ‘due’; steaks are actually fairly common; long term probabilities
don’t give us information on actual number of outcomes or the order in which they occurred.
Now for examples:
Use the following chart to answer the questions.
Male
Female
Total
Democrat
20
10
30
Republican
40
30
70
1. What is the probability of choosing a person that is female and Democrat?
2. What is the probability of choosing a person that is female or Democrat?
3. If a person is male, what is the probability that he is a democrat?
Class handouts
Total
60
40
100