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5.2 Probability RulesObjectives
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.”
There are 4 outcomes that result in a sum of 5. Since each outcome has probability 1/36, P(A) = 4/36.
Suppose event B is defined as “sum is not 5.” What is P(B)?
P(B) = 1 – 4/36 = 32/36
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
• 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.
•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,
•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).
Basic Probability Rules
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=.584orP(3 or better)=1-P(2 or worse)=1-(.233+.183)=1-.416=.584c) Find the probability that the chosen student didn’t get a 1.P(not 1)= 1-P(1)=1-.233=.767orP(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?
If A and B are any two events resulting from some chance process, thenP(A or B) = P(A) + P(B) – P(A and B)
General Addition Rule for Two Events
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 studenta) has pierced earsP(pierced ears) = 103/178b) is a male with pierced earsP(male and pierced ears) = 19/178c) is a male or has pierced earsP(male or pierced ears) = 90/178 + 103/178 – 19/178 = 174/178We 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 memberb) Is a high school graduateP(graduate) = 310/500c) Is a high school graduate and owns a homeP(graduate and homeowner) = 221/500d) Is a high school graduate or owns a homeP(graduate or homeowner) = 119/500 + 221/500 + 89/500 = 429/500or 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 = .98d) Find the probability that the household has neither type of phoneP(no land and no cell) = .02e) Find the probability the household has a cell phone only.P(only cell) = .38