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5.2 Probability Rules
Definition: Sample Space and 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. Definition: 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.
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Ex: Roll the Dice ­ building a probability model
P(A) = P(sum is 5) = 4/36
P(B) = P(sum is not 5) =P(not A) = 32/36 note that P(A) +P(B) = 1
P(C) = P(sum is 6) = 5/36
P(A or C) = P(sum of 5) + P(sum of 6) = 4/36 + 5/36 = 9/36
in other words P(A or C ) = P(A) + P(C)
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BASIC RULES OF PROBABILITY
1. The probability of any event is a number between 0 and 1
2. All possible outcomes together must have probabilities whose sum is 1.
3. 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 A
total number of outcomes in sample space
4. The probability that an event does not occur is 1 minus the probability that the event does occur.
We refer to "Not A" as the complement of A and
denote it as AC
5. If two events have no outcomes in common, the probability that one or the other occurs is the sum of their individual probabilities. Definition: mutually exclusive (disjoint)
Two events are mutually exclusive (disjoint) if they have no outcomes in common and so can never occur together.
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Symbolically Representing the Basic Probability Rules
1. For any event A, 0 ≤ P(A) ≤ 1
2. If S is the sample space in a probability model , P(S) = 1
3. In the case of equally likely outcomes, P(A) = number of outcomes corresponding to event A
total number of outcomes in sample space
4. Complement Rule: P( AC ) = 1 ­ P(A )
5. Addition Rule for mutually exclusive events: If A and B are mutually exclusive P(A or B) = P(A) + P (B )
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Example: Distance Learning p302
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Some Examples to help with the basic rules of probability.
1. Roll a Die Once (basic probability) S = { 1, 2, 3, 4, 5, 6}
Find the P(odd) = 3/6 = 1/2 = 50%
Find the P(# > 4 ) = 2/6 = 1/3 = 33.33%
2. Checking for Mutually Exclusive Events
Flip a Coin Twice
Define 3 Events
A = two heads {HH}
B = two tails {TT}
C = At least one head {HT, TH, HH}
Are A and B mutually exclusive? i.e. Do these events have any outcomes in common?
HH ≠ TT Nothing in common = Mutually Exclusive
Are A and C Mutually Exclusive?
HH = HH [have an outcome in common] ­ NO
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3. Mutually Excusive?
Roll a Die
A = Even # {2, 4, 6}
B = Odd #
{1, 3, 5}
C = # < 4
{1, 2, 3}
Are A and B mutually exclusive?
Yes no common outcome
Are B and C mutually exclusive? No 1 and 3 are common
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Complement Example
1. The probability that a randomly selected student attended at least 1 major league baseball game is .12. What is the probability that they do not attend a game all year?
A = attend game
Will also see the notation A
P( AC) = 1 ­ .12 = .88 or 88%
2. 20% of all coffee drinkers prefer black coffee. What is the probability that a randomly selected coffee drinker does not prefer black coffee?
P(Ac) = 1 ­ P(A) = 1 ­ .2 = .8 = 80%
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In a group of 200 adults, 39 have a college degree. If one adult is selected at random, what is the probability that the adult has a college degree?
P(college degree) = 39/200 = .195 or 19.5%
Suppose Phil randomly selects 1000 Calif. residents and observes that 540 of them own their home and 460 rent.
What is the probability that a randomly selected Calif. resident owns their home?
P(own) = 540/1000 = 54%
P(rent) = 460/1000 = 46% note: P(own) + P (rent) = 1 9
Example: Who has pierced Ears? p303
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TWO WAY TABLES AND PROBABILITY
Notes: "OR" ­ In statistics means one, the other, or both!!
Venn Diagrams ­ useful in Probability
­ diagram p. 304
Sample Space
Event A
Event B
Event A and B
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GENERAL ADDITION RULE FOR TWO EVENTS
If A and B are two events resulting from some chance process, then
P(A or B) = P(A) + P(B) ­ P (A and B)
If the events are mutually exclusive then P(A and B) =0
Union = the collection of outcomes that belong to A or B or to Both A and B
P (A ∪ B)
Intersection = The collection of all outcomes that are common to both A and B.
P (A ∩ B)
So the addition rule could be written
P (A ∪ B) = P(A) + P(B) ­ P (A ∩ B)
subtract out b/c counted twice!
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Example p. 306 Who has pierced ears?
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Example: p307 Who reads the paper?
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Example:
Coffee Drinkers
Cream
Male
60
Female 90
Total
150
Sugar
120
90
210
Black
60
30
90
Total
240
210
450
1. What is the probability that a randomly selected coffee drinker is a "Female" or "Prefers Sugar"?
P(F∪S) = P(F) + P (S ) ­ P(F∩S)
= 210/450 + 210/450 ­ 90/450 = 330/450 = .73333
2. What is the probability that a randomly selected coffee drinker is Male or Prefers Black Coffee?
P(M∪B) = P(M) + P (B ) ­ P(M∩B)
= 240/450 + 90/450 ­ 60/450 = 270/450 = .6 = 60%
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Example: Basketball Players at a college graduating
Male
Female
Total
Grad
126
133
259
Not Grad
55
32
87
Total
181
165
346
What is the probability that a randomly selected player is Female or Graduated?
P(F∪G) = P(F) +P(G) ­ P(F∩G) = 165/346+ 259/346 ­ 133/346 = .8410 = 84.10%
What is the probability that a randomly selected player is Male or did Not Graduate?
P(M∪G) = P(M) +P(G) ­ P(M∩G) = = 181/346 + 87/346 ­ 55/346 = .6156 = 61.56%
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Example: An outdoor wedding is being planned. The prob. of rain is .2 and the prob. of a fight is .2. The prob. of both events happening is .16. What is the prob. that it will rain or a fight will break out at the wedding?
P(R∪F) = P(R) +P(F) ­ P(R∩F) = .2 +.2 ­ .16 = .24 or 24%
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Homework:
p298 27, 31,32 p309 43­47odd
p298 + p309
29, 33­36, 49­55odd
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