Download probability

Probability Models
Definitions
probability model – describes all possible outcomes and says how to assign probabilities to
any collection of outcomes
sample space – collection of unique outcomes of a random circumstance
– we usually denote this by S
event – a collection of outcomes
Suppose you are asked to toss a coin. What is the sample space?
You toss the coin and it lands on Tails. What is the outcome?
If you toss the coin three times and it lands THH then THH is the event.
Probability Rules
Rule 1. Any probability is a number between 0 and 1. So if we observe an event A then we
know 0  P(A)  1
Rule 2. All possible (disjoint or non-overlapping) outcomes together must have probability 1.
• An outcome must occur on every trial.
• The sum of the probabilities for all possible outcomes must be exactly 1.
Consider the following assignment of probabilities
Marital Status of a Random Sample of Women
Ages 25 to 29
Marital Status
Probability
Never married
0.386
Married
0.555
Widowed
0.004
Divorced
0.055
• Each of the probabilities is a number between 0
and 1.
• The probabilities total to 1.
0.386 + 0.555 + 0.004 + 0.055 = 1
• Any assignment of probabilities to all individual
outcomes that satisfies Rules 1 and 2 is
legitimate.
Rule 3. The probability that an event does not occur is 1 minus the probability that the event
does occur. This is known as the complement rule.
• Suppose that P(A) = .70
• Using this rule we can determine P(“not” A). The event “not A” is known as
the complement of A which can be written as AC
P(“not” A) =
Clicker Questions about Complements and their probabilities
Suppose the probability of a horse winning a race is 0.85. What is the complement of the
horse winning?
A. Horse winning
B. Horse not winning
C. 0.85
D. 0.15
What is the probability of the horse not winning?
A. Horse winning
B. Horse not winning
C. 0.85
D. 0.15
Rule 4. Union Rule: If two events have no outcomes in common, the probability that one or
the other occurs is the sum of their individual probabilities. If this is true then the events are
said to be disjoint.
Suppose events A and B are disjoint and you know that P(A) = .40 and P(B) = .35.
What is the P(A or B)?
A Venn diagram can help us picture events and sample spaces.
Chapter 18
Page 2
Example illustrating more general form of Rule 4
The probability a student is in honors math is 0.25, the probability a student is in honors
science is 0.3, and the probability a student is in both is 0.2.
What is the probability a student is in at least one honors class?
This is called a union and can be pictured on a Venn diagram
Probability of honors math or science (or both) =
P(honors math or honors science)
The probability it will rain Wednesday AM is 30%. The probability it will rain Wednesday PM is
30%. The probability it will rain both Wednesday AM and Wednesday PM is 10%.
What is the probability it will rain on Wednesday?
A) 20%
D) 50%
B) 30%
E) 60%
C) 40%
Rule 5. Multiplication Rule: If two events, A and B, are independent then P(A and B) =
P(A)P(B)
Independence means that the occurrence of event A does not affect the occurrence of event B
What is the probability that the first roll of a die is even, and the second roll is odd?
A – 1/36
B – 1/12
C – 1/4
D – 1/2
Chapter 18
Page 3
̂
Sampling Distribution of 𝒑
sampling distribution – tells what values a statistic takes in repeated samples from the same
population
• We can’t predict the outcome of one sample, but the outcomes of many samples have a
regular pattern.
• The distribution of the statistic tells us what values it can take and how often it takes
those values
Take an SRS of 1523 adults. Ask each if they bought a
lottery ticket in the last 12 months.
The sample proportion who say “yes” is
Suppose the true proportion, p = 0.60. If we were to
take 1000 SRSs and calculate 1000 𝑝̂ ’s the distribution
would look like the picture to the right….
Example
An opinion poll asks an SRS of 501 teens, “Generally speaking, do you approve or
disapprove of legal gambling or betting?” Suppose that in fact exactly 50% of all teens
would approve if asked (so p = 0.5).
Suppose the sample proportion, 𝑝̂ , who approve is normally distributed with mean 0.5 and
standard deviation 0.022.
What is the probability that a poll will indicate less than 47.8% of teens approve?
Chapter 18
Page 4