5.2 Probability Rules
Objectives
SWBAT:
• DESCRIBE a probability model for a chance process.
• USE basic probability rules, including the complement rule
and the addition rule for mutually exclusive events.
• USE a two-way table or Venn diagram to MODEL a chance
process and CALCULATE probabilities involving two events.
• USE the general addition rule to CALCULATE probabilities.
What is a sample space? (not to be confused with blank space…if it was a blank space you’d
write your name)
What is a probability model?
The sample space S of a chance process is the set of all possible outcomes.
A probability model is a description of some chance process that consists of two parts:
• a sample space S and
• a probability for each outcome.
• Let’s say we toss a coin one time. There are only two possible outcomes: heads and tails.
• We write the sample space using set notation as S={H,T}.
• The probability for each of these outcomes is 0.50.
What is an event?
An event is any collection of outcomes from some chance process. That is, an event is a subset of
the sample space. Events are usually designated by capital letters, like A, B, C, and so on.
If A is any event, we write its probability as P(A).
Let’s say we roll a pair of dice. Here is our sample space:
Suppose we define event A as “sum is
5.”
Suppose event B is defined as
“sum is not 5.” What is P(B)?
P(B) = 1 – 4/36 = 32/36
There are 4 outcomes that result in a sum of 5.
Since each outcome has probability 1/36, P(A) = 4/36.
Imagine flipping a fair coin three times. Describe the probability model for this
chance process and use it to find the probability of getting at least 1 head in
three flips.
• S = {HHH, HHT, HTH, HTT, TTT, TTH, THT, THH}
• Each of these 8 outcomes will be equally likely and have a probability of .125.
• P(at least 1 head) = 7/8 = .875.
Summarize the five basic probability rules.
What does it mean if two events are mutually exclusive?
• The probability of any event is a number between 0 and 1.
• All possible outcomes together must have probabilities whose sum is exactly 1.
• If all outcomes in the sample space are equally likely, the probability that event A occurs can be
found using the formula
P(A) =
number of outcomes corresponding to event A
total number of outcomes in sample space
• The probability that an event does not occur is 1 minus the probability that the event does
occur.
• If two events have no outcomes in common, the probability that one or the other occurs is the
sum of their individual probabilities.
Two events A and B are mutually exclusive (disjoint) if they have no outcomes in common and
so can never occur together—that is, if P(A and B ) = 0.
Example: If we randomly select a student from LHS, what is the probability that the student is both a
sophomore and a senior? 0!
We can summarize the basic probability rules more concisely in symbolic form.
Basic Probability Rules
•For any event A, 0 ≤ P(A) ≤ 1.
•If S is the sample space in a probability model,
P(S) = 1.
•In the case of equally likely outcomes,
number of outcomes corresponding to event A
P(A) =
total number of outcomes in sample space
•Complement rule: P(AC) = 1 – P(A)
•Addition rule for mutually exclusive events: If A and B are mutually exclusive,
P(A or B) = P(A) + P(B).
Randomly select a student who took the 2010 AP Statistics exam and record the student’s score.
Here is the probability model:
a) Show this that is a legitimate probability model.
All the probabilities are between 0 and 1 and the sum of the probabilities is 1, so
this is a legitimate probability model.
b) Find the probability that the chosen student scored 3 or better.
P(3 or better)= .235+.224+.125=.584
or
P(3 or better)=1-P(2 or worse)=1-(.233+.183)=1-.416=.584
c) Find the probability that the chosen student didn’t get a 1.
P(not 1)= 1-P(1)=1-.233=.767
or
P(not 1)= .183+.235+.224+.125=.767
What is the general addition rule? Is it on the formula sheet? What if the events are
mutually exclusive?
General Addition Rule for Two Events
If A and B are any two events resulting from some chance process, then
P(A or B) = P(A) + P(B) – P(A and B)
When finding probabilities involving two
events, a two-way table can display the
sample space in a way that makes
probability calculations easier.
Suppose we choose a student at random. Find the probability that the
student
a) has pierced ears
P(pierced ears) = 103/178
b) is a male with pierced ears
P(male and pierced ears) = 19/178
c) is a male or has pierced ears
P(male or pierced ears) = 90/178 + 103/178 – 19/178 = 174/178
We need to subtract the 19 so that we do not double count it.
Alternative method:
P(male or pierced ears) = 71/178 + 19/178 + 84/178 = 174/178
On the formula sheet:
Who owns a home?
What is the relationship between educational achievement and home ownership? A random
sample of 500 U.S. adults was selected. Each member of the sample was identified as a high
school graduate (or not) and as a home owner (or not). Overall, 340 were homeowners, 310
were high school graduates, and 221 were both homeowners and high school graduates.
a) Create a two-way table for the data.
Suppose we chose a member of the sample at
random. Find the probability that the member
b) Is a high school graduate
P(graduate) = 310/500
c) Is a high school graduate and owns a home
P(graduate and homeowner) = 221/500
d) Is a high school graduate or owns a home
P(graduate or homeowner) = 119/500 +
221/500 + 89/500 = 429/500
or P(graduate or homework) = 310/500 +
340/500 – 221/500 = 429/500
Note: Show work. You must
AT LEAST show the fraction.
According to the National Center for Health Statistics, in December 2012, 60% of US
households had a traditional landline telephone, 89% of households had cell phones, and 51%
had both. Suppose we randomly selected a household in December 2012.
a) Make a two-way table to displays the sample space of this
chance process.
b) Construct a Venn diagram to represent the outcomes of this
chance process.
c) Find the probability that the household has at least one of the
two types of phones
P(cell or land) = .09+.51+.38 = .98
d) Find the probability that the household has neither type of
phone
P(no land and no cell) = .02
e) Find the probability the household has a cell phone only.
P(only cell) = .38