unit 7: probability. 7.1: terminology i’m going to roll a six-sided die. rolling a die is called...
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7.1: Empirical vs. Theoretical Two kinds of probability: Empirical: based on observation of experiments Theoretical: what should happenTRANSCRIPT
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 numberThis event includes the outcomes 2, 4, 6
7.1: Empirical vs. Theoretical
Two kinds of probability:Empirical: based on observation of experimentsTheoretical: what should happen
7.1: Finding Probability
number of times event has occurredtotal 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 Problemp275 #17
AnimalAnimal Number Number TreatedTreated
Dog 45Cat 40Bird 15
Rabbit 5TOTAL 105
P(next animal is a dog) =45
105
7.2: Theoretical Probability
number of favorable outcomes
total number of possible outcomes
7.2: Important FactsProbability of an event that can’t happen is 0Probability of an even that must happen is 1Every 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) = 452
Can you reduce the fraction?
7.2: Example 3(b)
P (drawing NOT a 5) = 5248 =
1312
NOTICE!
P (drawing NOT a 5) = 1 - P (drawing a 5)
7.2: Practice Problemsp 282 #21, 23
P(drawing a black card) =
P(drawing a red card or a black card) =
7.2: Practice Problemsp282 #27
P (red) =
P (green) =
P (yellow) =
P (blue) =
4
4
4
=
7.3: Odds
Odds against an event = P(success)
P(failure)
Odds in favor of an event = P(success)P(failure)
7.3: Practice Problemsp 291 #53
Odds against selling out = P(sells out)P(does not sell out)
Odds against selling out = 1 - 0.9
0.9= 0.11
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 Problemp301 #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 Problemp301 #57
(b) Expected Value =
$1*0.5625 + $10*0.25 + $20*0.125 + $100*0.0625
= $11.8125
7.4: Practice Problemp301 #57
(c) Fair Price = Expected Value + Cost0 = $11.8125 + CC = $11.8125
(it makes sense to round to two decimal places)
7.5: Tree DiagramsCounting 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 Problemp311 #11 (a) 2*2 = 4
p311 #11 (b)
H
T
H
THT
Sample Space
HHHTTHTT
7.5 Practice Problem
p311 #11 (c) 1/2p311 #11 (d) 2/4 = 1/2p311 #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 Problemp323 #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 Problemp323 #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