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The probable is what usually
happens. Aristotle
Chapter 6: Probability
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Chance experiment: any activity or
situation in which there is uncertainty
about which outcome will result
Fate laughs at probabilities. Bulwer, Lytton E.G.
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Sample Space: collection of all
possible outcomes of a chance experiment
Roll a dieS={1,2,3,4,5,6}
Probability is the very guide of life. Cicero
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What is the sample space?
1.) Rolling 1 die
2.) Rolling 2 dice and counting the number of pips face up
3.) Toss a coin
4.) Toss a coin three times and record the results
5.) Flip a coin and roll a die
S = {1,2,3,4,5,6}
S = {2,3,4,5,6,7,8,9,10,11,12}
S = {H,T}
S = {HHH,HHT,HTH,THH,HTT,THT,TTH,TTT}
S = {H1,H2,H3,H4,H5,H6,T1,T2,T3,T4,T5,T6}
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An event is any collection of outcomes from the sample space of a chance experiment.
– Rolling a prime # E = {2,3,5}
A simple event is an event consisting of exactly one outcome.
Rolling a 6: E = {6}
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Venn Diagrams•Used to display relationships
•between events
•Helpful in calculating probabilities
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Complement •Consists of all outcomes that are not in the event
–Not rolling a even #EC={1,3,5}
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The rectangle represents the sample space and shaded area represents the event A.
The shaded area represents the event not A.
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Union •the event A or B happening•consists of all outcomes that are in at least one of the two events–Rolling a prime # or even number
E={2,3,4,5,6}
BAE
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The shaded area represents the event
A or B ( the union of two events).
Union (or)
BAE
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Intersection •the event A and B happening
•consists of all outcomes that are in both events–Drawing a red card and a “2” – E = {2 hearts, 2 diamonds}
BAE
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Intersection (and)
The shaded area represents the event
A and B (the intersection of two sets).
BAE
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Mutually Exclusive (disjoint)
•two events have no outcomes in common
–Roll a “2” or a “5”
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• Two events that have no common outcomes are said to be disjoint or mutually exclusive.
A and B are disjoint events
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Some illustrations
A B
C
A B C
A B
C
A, B & C are Disjoint
A B
C
A B C
A B
C
A B Cc
A ∩ B ∩ C
A υ B υ C A ∩ B ∩ Cc
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Com Sci
Statistics & Computer Science & not Calculus
Stat Cal
Com Sci
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Calculus or Computer Science
Stat Cal
Com Sci
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(Statistics or Computer Science) and not Calculus
Stat Cal
Com Sci
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Statistics and not (Computer Science or Calculus)
Stat Cal
Com Sci
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Stat Cal
Com Sci
Statistics or Computer Science?
30 10520
1010
20
80
170
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Stat Cal
Com Sci
Statistics and Computer Science?
30 10520
1010
20
80
20
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Stat Cal
Com Sci
Statistics or (Computer Science and Calculus)?
30 20 105
10 20
80
10
90
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Stat Cal
Com Sci
(Statistics or Computer Science) and Calculus?
30 20 105
10
10
20
80
50
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Country-Western songs emphasize three basic themes: love, prison, and trucks. A survey of the local Country-Western radio station produced the following data: 12 songs about a truck driver who is in love while in prison2 about people in prison who are not in love and do not drive trucks8 about people who are out of prison, are not in love, and do not drive a truck13 about a prisoner in love28 about a person in love18 about a truck driver in love16 about truck drivers who are not in prison3 about a truck driver in prison who is not in love
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12 songs about a truck driver who is in love while in prison2 about people in prison who are not in love and do not drive trucks8 about people who are out of prison, are not in love, and do not drive a truck13 about a prisoner in love28 about a person in love18 about a truck driver in love16 about truck drivers who are not in prison3 about a truck driver in prison who is not in love
How many songs were about:a.) truck drivers?
b.) prisoners?
c.) truck drivers in prison?
d.) people not in prison?
e.) people not in love?
f.) How many songs were surveyed?
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Always be a little
improbable.Wilde, Oscar