Transcript Document

Unit 7: Probability
7.1: Terminology
I’m going to roll a six-sided die.
Rolling a die is called an “experiment”
The number I roll is called an “outcome”
An “event” is a group of outcomes
Example of event: roll an even number
This event includes the outcomes 2, 4, 6
7.1: Empirical vs.
Theoretical
Two kinds of probability:
Empirical: based on observation of
experiments
Theoretical: what should happen
7.1: Finding Probability
number of times event has occurred
total number of experiments done
7.1: Law of Large
Numbers
Law of Large Numbers: Probability applies to a
large number of trials, not a single experiment.
Ex: Baby gender. The probability of having
a boy is 50%. My sister-in-law just had a
girl (and is expecting another!). The
probability works when applied to the whole
population (large number of trials), not when
applied to my sister-in-law (a single
experiment).
7.1: Practice Problem
p275 #17
Number
P(next animal is a dog) =
Dog
45
45
Cat
40
105
Bird
15
Rabbit
5
TOTAL
105
Animal Treated
7.2: Theoretical
Probability
number of favorable outcomes
total number of possible outcomes
7.2: Important Facts
Probability of an event that can’t happen is 0
Probability of an even that must happen is 1
Every probability is a number from 0 to 1.
The sum of all probabilities for an experiment
is 1.
7.2: Example 3 (a)
P(drawing a 5) =
4
52
Can you reduce the fraction?
7.2: Example 3(b)
P (drawing NOT a 5) =
48
52
=
12
13
NOTICE!
P (drawing NOT a 5) = 1 - P (drawing a 5)
7.2: Practice Problems
p 282 #21, 23
P(drawing a black card) =
P(drawing a red card or a black card) =
7.2: Practice Problems
p282 #27
P (red) =
=
4
P (green) =
4
P (yellow) =
4
P (blue) =
7.3: Odds
P(failure)
Odds against an event =
P(success)
Odds in favor of an event =
P(success)
P(failure)
7.3: Practice Problems
p 291 #53
Odds against selling out =
P(does not sell out)
Odds against selling out =
= 0.11
P(sells out)
1 - 0.9
0.9
7.4: Expected Value
To find expected value:
For each outcome, multiply the probability
times the value of that outcome.
Add the results together for all possible
outcomes.
7.4: Fair Price
Fair Price = Expected Value + Cost to Play
7.4: Practice Problem
p301 #57
(a) P(1) = 9/16 = 0.5625
P(10) = 4/16 = 0.25
P(20) = 2/16 = 0.125
P(100) = 1/16 = 0.0625
7.4: Practice Problem
p301 #57
(b) Expected Value =
$1*0.5625 + $10*0.25 + $20*0.125 + $100*0.0625
= $11.8125
7.4: Practice Problem
p301 #57
(c) Fair Price = Expected Value + Cost
0 = $11.8125 + C
C = $11.8125
(it makes sense to round to two decimal places)
7.5: Tree Diagrams
Counting Principal: If there are M possible
outcomes for a first experiment and N possible
outcomes for a second experiment, there are
M*N total possible outcomes.
Ex: I have three shirts and two pairs of pants.
I can make 3*2 = 6 outfits.
The list of possible outcomes (outfits) is the
“sample space”
7.5: Practice Problem
p311 #11 (a) 2*2 = 4
Sample Space
p311 #11 (b)
H
H
T
T
H
T
HH
HT
TH
TT
7.5 Practice Problem
p311 #11 (c) 1/2
p311 #11 (d) 2/4 = 1/2
p311 #11 (e) 1/2
7.6: Or and And
Problems
P(A or B) = P(A) + P(B) - P(A and B)
P(A and B) = P(A) * P(B)
Mutually Exclusive: P(A and B) = 0
7.6: Example 1
P(Even or >6) = P(Even) + P(>6) - P(Even and >6) =
5/10 + 4/10 - 2/10 = 7/10
7.6: Practice Problem
p323 #97
(a) No. The probability of the second event is affected
by the outcome of the first.
(b) 0.001
(c) P(A and B) = P(A)*P(B) = 0.001 * 0.04 = 0.00004
7.6: Practice Problem
p323 #97
(d) P(A and NOT B) = P(A)*P(NOT B) = 0.001*0.96 =
0.00096
(e) P(NOT A and B) = P(NOT A)*P(B) = 0.999*0.001 =
0.000999
(f) P(NOT A and NOT B) = P(NOT A)*P(NOT B) =
0.999*0.999 = 0.998001