Probability and Probability and Simulation Simulation Rules in ProbabilityRules in Probability

Probability RulesProbability Rules1. Any probability is a number between 0 and 11. Any probability is a number between 0 and 1 0 ≤ P[A] ≤ 10 ≤ P[A] ≤ 12. The sum of all the probabilities of all possible 2. The sum of all the probabilities of all possible outcomes must equal 1.outcomes must equal 1. P[S] = 1P[S] = 13. If two events have no outcomes in common, 3. If two events have no outcomes in common, the probability that the one or the other occurs the probability that the one or the other occurs is the sum of their individual probabilities.is the sum of their individual probabilities. P[A or B] = P[A] + P[B]P[A or B] = P[A] + P[B]4. The probability that an event does not occur 4. The probability that an event does not occur is 1 minus the probability that the event does is 1 minus the probability that the event does occur.occur. P[A’] = 1 - P(A)P[A’] = 1 - P(A)

Probability Rules Probability Rules (Examples)(Examples)

1. 0 ≤ P[A] ≤ 1:1. 0 ≤ P[A] ≤ 1: any proportion is always between any proportion is always between 0 and 10 and 1

2. P[S] = 1 :2. P[S] = 1 : probability of getting 1, 2, 3, 4, 5, probability of getting 1, 2, 3, 4, 5, or 6 in tossing a die is ⅙+⅙+⅙+⅙+⅙+⅙= 6/6 or or 6 in tossing a die is ⅙+⅙+⅙+⅙+⅙+⅙= 6/6 or 11

3. P[A or B] = P[A] + P[B]: 3. P[A or B] = P[A] + P[B]: tossing a coin 3 times A: tossing a coin 3 times A: getting all tails B: getting all heads P[A or B] = ⅛ getting all tails B: getting all heads P[A or B] = ⅛ + ⅛ = 2/4 or ¼ + ⅛ = 2/4 or ¼

4. P[A’] = 1 - P(A): 4. P[A’] = 1 - P(A): Probability of getting A in AP Probability of getting A in AP Stat is 20% therefore, the probability of NOT Stat is 20% therefore, the probability of NOT getting an A in AP Stats is 80%getting an A in AP Stats is 80%

Draw a woman aged 25 - 34 years old at Draw a woman aged 25 - 34 years old at random and record her marital status. If we random and record her marital status. If we drew many women, this is the proportion we drew many women, this is the proportion we would get. Here’s the probability model:would get. Here’s the probability model:

Marital Marital StatusStatus

Never Never MarriedMarried MarriedMarried WidowedWidowed DivorcedDivorced

ProbabilitProbabilityy

0.2980.298 0.6220.622 0.0050.005 0.0750.075

P(not married)P(not married) = 1-P(married)= 1-P(married)= 1 - 0.622= 1 - 0.622= 0.378= 0.378

Therefore there are Therefore there are 37.8%37.8% of women that are NOT of women that are NOT marriedmarried

Marital Marital StatusStatus

Never Never MarriedMarried MarriedMarried WidowedWidowed DivorcedDivorced

ProbabilitProbabilityy

0.2980.298 0.6220.622 0.0050.005 0.0750.075

P(never married or P(never married or divorced)divorced)

= P(never married) + P(divorced)= P(never married) + P(divorced)

= 0.298 +0.075 = 0.298 +0.075

= 0.373= 0.373

Therefore there are Therefore there are 37.3%37.3% of women in of women in this group that are either never married this group that are either never married

or divorcedor divorced

Venn DiagramVenn Diagram

SS

AA

BB A’A’

AA

Disjoint events A Disjoint events A

and B (and B (mutually mutually

exclusive eventsexclusive events))

A’ = complement of A’ = complement of AA

P[A or B ] = P[A] + P[B] P[A or B ] = P[A] + P[B] P[A and B ] = P[A] x P[A and B ] = P[A] x P[B] P[B]

Addition Rule of Addition Rule of union eventunion event

Multiplication Rule Multiplication Rule of Independent of Independent

VariablesVariablesEx: Probability of Ex: Probability of getting a sum of 3 getting a sum of 3 and probability of and probability of getting a sum of 5 getting a sum of 5

when you roll a pair when you roll a pair of diceof dice

Ex: Probability of Ex: Probability of getting getting a getting getting a

4 and then 4 and then another 4 when another 4 when

you roll a die you roll a die twicetwice

P[A or B] = 2/36 + P[A or B] = 2/36 + 4/36 4/36 = 6/36= 6/36= 17%= 17%

P[A and B] = 1/6 x P[A and B] = 1/6 x 1/61/6= 1/36= 1/36

= 2.8%= 2.8%

P [A P [A ∩∩ B]B]

P [A P [A ∪∪ B] B]