Terms
PROBABILITY
Important Random Experiment a process leading to an uncertain outcome Basic Outcome a possible outcome of a random experiment Sample Space the collection of all possible outcomes of a random experiment Event any subset of basic outcomes from the sample space
Coin: 2 outcomes S = {H, T}
Die: Six outcomes S = {1,2,3,4,5,6}
Cards: 52 outcomes S = {13 of clubs,13 of spades,13 of hearts,13 of diamonds}
Examples:
Let the Sample Space be the collection of all possible outcomes of rolling one die:
S = [1, 2, 3, 4, 5, 6] Let A be the event "Number rolled is even" Let B be the event "Number rolled is at least 4" Then A = [2, 4, 6] and B = [4, 5, 6]
Important Terms:
S = [1, 2, 3, 4, 5, 6] A = [2, 4, 6] B = [4, 5, 6]
Complement:
A C = [1, 3, 5]
Intersection: A and B
BC = [1, 2, 3]
A ! B = [4, 6]
Union: A or B
A ! B = [5]
C
A ! B = [2, 4, 5, 6]
C
A!A
= [1, 2, 3, 4, 5, 6] = S
Probability
Probability the chance that an uncertain event will occur
P(Heads) = 1/2
P(get a 5) = 1/6 P(get an even number) = 3/6 = 1/2
Equally Likely Outcomes
When outcomes are equally likely, probabilities for events are easy to find just by counting: It's equally likely to get heads or tails from the toss of a fair coin. It's equally likely to get any one of six outcomes from the roll of a fair die.
However, keep in mind that events are not always equally likely: A skilled basketball player has a better than 50-50 chance of making a free throw.
Classical Definitions of Probability
number of outcomes in A m P(A) = = total number of all possible outcomes n
True for equally likely outcomes!
Probability
Probability the chance that an uncertain event will occur - always takes the value between 0 and 1
0 P(A) 1 For any event A
P(A) = 0 A - Impossible event P(A) = 1 A - Certain event 0
Impossible
1
Certain
.5
The French naturalist Count Buffon (1707-1788) tossed a coin 4040 times. Result: 2048 heads. Proportion: 2048/4040=0.5069
Around 1900, the English statistician Karl Pearson tossed a coin 24,000 times. Result: 12,012 heads. Proportion: 12012/24000= 0.5005
The Law of Large Numbers (LLN)
The long-run relative frequency of repeated independent trials gets closer and closer to the true value of probability as the number of trials increases.
For example, consider flipping a fair coin many, many times. The overall percentage of heads should settle down to about 50% as the number of outcomes increases.
Statistical Definitions of Probability
number of trials A occurred f P(A) = = total number of repeated trials n
(relative frequency as probability)
3. Personal Probability
A personal probability expresses an individual's judgment of how likely the outcome is.
More than one event...
The First Three Rules for Working with Probability Rules
1. Make a picture.
2. Make a picture.
3. Make a picture.
Picturing Probabilities
The most common kind of picture to make is called a Venn diagram.
British philosopher and mathematician John Venn (18341923) introduced the Venn diagram in 1881
Venn diagrams
The probability of the set of all possible outcomes of an experiment must be 1
P(S) = 1
(S represents the set of all possible outcomes.)
Venn diagrams
The set of outcomes that are not in the event A is called the complement of A, denoted AC The probability of an event occurring is 1 minus the probability that it doesn't occur
P(A) = 1 P(AC)
Venn diagrams
Events that have no outcomes in common and, thus, cannot together occur are called mutually exclusive (or disjoint).
Examples:
S = [1, 2, 3, 4, 5, 6] A = [2, 4, 6] B = [4, 5, 6] C = [1, 2]
Mutually exclusive:
A and B are not mutually exclusive: the outcomes 4 and 6 are common to both B and C are mutually exclusive: there are no common outcomes
Examples:
S = [1, 2, 3, 4, 5, 6] A = [2, 4, 6] B = [4, 5, 6] C = [1, 2]
Intersection A and B are not mutually exclusive: the outcomes 4 and 6 are common to both A and B = [4, 6]
Probability Rules: Addition Rule:
For two mutually exclusive events A and B, the probability that one or the other occurs is the sum of the probabilities of the two events. P(A or B) = P(A) +P(B)
Probability Rules: General Addition Rule
For any two events A and B, the probability that one or the other occurs P(A or B) is the sum of the probabilities of the two events minus the probability of the intersection (A and B).
P(A or B) = P(A) + P(B) P(A and B)
P(A or B)=?
Are A and B YES mutually exclusive events? NO
P(A or B) = P(A) +P(B)
P(A or B) = P(A) + P(B) P(A and B)
Addition Rule Example
Consider a standard deck of 52 cards, with four suits:
Let event A = card is an Ace Let event B = card is from a red suit
A and B are not mutually exclusive events: two red aces
Addition Rule: Example
P(Red or Ace) = P(Red) + P(Ace) - P(Red and Ace) = 26/52 + 4/52 - 2/52 = 28/52
Don't count the two red aces twice!
Type Ace Non-Ace Total
Color
Red Black
Total 4 48 52
2 24 26
2 24 26
Independency
Two events A and B are independent if the fact that A occurs does not affect the probability of B occurring
With replacement
Without replacement
A probability that takes into account a given condition is called conditional probability P(B|A) "the probability of B given A".
Probability Rules: Multiplication Rule:
For two independent events A and B, the probability that both A and B occur is the product of the probabilities of the two events.
P(A and B) = P(A) x P(B)
Multiplication Rule may be extended to three or more independent events:
P(A and B and C and D and E) = = P(A) x P(B) x P(C) x P(D) x P(E)
Probability Rules: General Multiplication Rule
For any two events A and B, the probability that both A and B occur is the product of the probability of first event and the probability of the second, given that the first event occurred. For any two events A and B,
P(A and B) = P(A) x P(B|A)
or
P(A and B) = P(B) x P(A|B)
P(A and B)=?
Are A and B YES independent?
P(A and B) = P(A) x P(B)
NO
P(A and B) = P(A) x P(B|A)
Conditional Probability Formula:
P(A and B) P(B A) = and P(A)
For independent events
P(A and B) P(A B) = P(B)
P(A B) = P(A)
P(B A) = P(B)
and P(A and B) = P(A) ! P(B)
Events A and B are independent whenever P(B|A) = P(B).
Equivalently, events A and B are independent whenever P(A|B) = P(A).
Conditional Probability and Contingency Table
High blood pressure Men Women Total 17 20 37
Normal blood pressure 70 68 138
Total
87 88 175
Independent Mutually exclusive
Disjoint events cannot be independent! Well, why not?
Since we know that disjoint events have no outcomes in common, knowing that one occurred means the other didn't. Thus, the probability of the second occurring changed based on our knowledge that the first occurred. It follows, then, that the two events are not independent.
A common error is to treat disjoint events as if they were independent, and apply the Multiplication Rule for independent events--don't make that mistake.