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