Key Concepts. Key Concepts. Event Relations. Event Relations

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Eustace Freeman
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Probability and Probability Distributions Event Relations S B B Event Relations The intersection of two events, and B, is the event that both and B occur when the experient is perfored. We write B.

1 Probability and Probability Distributions Event Relations S B B Event Relations The intersection of two events, and B, is the event that both and B occur when the experient is perfored. We write B. S Event Relations The copleent of an event consists of all outcoes of the experient that do not result in event. We write C. B B C If two events and B are utually exclusive (disjoint), then B = Ø.

Select a student fro the classroo and record his/her hair color and gender.

C : B C: B C: Calculating Probabilities for Unions and Copleents B : dditive Rule : Suppose that there were 0 students in the

60/0 P( B) = P() + P(B) P( B) = Check: P( B) = Not Brown Special Case : ale with brown hair P() = 0/0 B: feale with brown hair

Probabilities for Copleents Since and C are disjoint, P( C )=P()+P( C ) lso, C = S, and P(S)=.

2 Select a student fro the classroo and record his/her hair color and gender. : student has brown hair B: student is feale C: student is ale What is the relationship between events B and C? C : B C: B C: Calculating Probabilities for Unions and Copleents B : dditive Rule : Suppose that there were 0 students in the classroo, and that they could be classified as follows: : brown hair Brown P() = 50/0 Male 0 40 B: feale Feale P(B) = 60/0 P( B) = P() + P(B) P( B) = Check: P( B) = Not Brown Special Case : ale with brown hair P() = 0/0 B: feale with brown hair P(B) = 30/0 and B are utually exclusive, so that Brown Male 0 40 Feale Not Brown P( B) = P() + P(B) = Calculating Probabilities for Copleents Since and C are disjoint, P( C )=P()+P( C ) lso, C = S, and P(S)=. Therefore, = P()+P( C ) P( C ) = P() C SIMPLE BUT VERY USEFUL! Select a student at rando fro the classroo. Define: : ale P() = 60/0 B: feale and B are copleentary, so that Brown Male 0 40 Feale Not Brown P(B) = - P() = - 60/0 = 60/0

Calculating Probabilities for Intersections Definition of Independence Foral definition: Two events are independent if P( Β)=P()P(B) If two events are NOT independent, then they are dependent. WHY?

P( B)=P( Β)/P(B) =[P()P(B)]/P(B) =[P()P(B)]/P(B) =P() Why? bowl contains five M&Ms, two red and three blue. Randoly select two candies, and define : second candy is red. B: first candy is blue.

3 Calculating Probabilities for Intersections Definition of Independence Foral definition: Two events are independent if P( Β)=P()P(B) If two events are NOT independent, then they are dependent. WHY? Conditional Probabilities given Defining Independence The first definition is really saying that if and B are independent, then P( B)=P() and P(B )=P(B). P( B)=P( Β)/P(B) =[P()P(B)]/P(B) =[P()P(B)]/P(B) =P() Why? bowl contains five M&Ms, two red and three blue. Randoly select two candies, and define : second candy is red. B: first candy is blue. re and B independent events? In a certain population, 0% of the people can be classified as being high risk for a heart attack. Three people are randoly selected fro this population. What is the probability that exactly one of the three are high risk? Suppose that the three people are selected one by one, and each one is classified in order of selection. What is the probability that the first high risk person is the third one selected in the saple? 3

Suppose we have additional inforation in the previous exaple. We know that only 49% of the population are feale. lso, of the feale patients, 8% are high risk. single person is selected at rando.

Then the probability of another event can be written as P() = P( S ) + P( S ) + + P( S k ) = P(S )P( S ) + P(S )P( S ) + + P(S k )P( S k ) The Law of Total Probability Bayes Rule S S S 3 Let S, S, S

4 Suppose we have additional inforation in the previous exaple. We know that only 49% of the population are feale. lso, of the feale patients, 8% are high risk. single person is selected at rando. What is the probability that it is a high risk feale? The Law of Total Probability Let S, S, S 3,..., S k be utually exclusive and exhaustive events (that is, one and only one ust happen). Then the probability of another event can be written as P() = P( S ) + P( S ) + + P( S k ) = P(S )P( S ) + P(S )P( S ) + + P(S k )P( S k ) The Law of Total Probability Bayes Rule S S S 3 Let S, S, S 3,..., S k be utually exclusive and exhaustive events with prior probabilities P(S ), P(S ),,P(S k ). If an event occurs, the posterior probability of S i, given that occurred is S S 3 S P() = P( S ) + P( S ) + P( S 3 ) = P(S )P( S ) + P(S )P( S ) + P(S 3 )P( S 3 ) Rando Variables Fro a previous exaple, we know that 49% of the population are feale. Of the feale patients, 8% are high risk for heart attack, while % of the ale patients are high risk. single person is selected at rando and found to be high risk. What is the probability that it is a ale? 4

Probability Distributions for Discrete Rando Variables We ust have 0 p(x i ) and p(x i ) = i Toss a fair coin three ties and define X =

HHH HHT HTH THH HTT THT TTH TTT X 3 0 P(X = 0) = P(X = ) = 3/8 P(X = ) = 3/8 P(X = 3) = x p(x) 0 3/8 3/8 3 Probability Histogra for x

Then the ean, variance and standard deviation of X are given as Mean : µ = x i p(x i ) i Variance : σ = (x i µ) p(x i ) Standard

5 Probability Distributions for Discrete Rando Variables We ust have 0 p(x i ) and p(x i ) = i Toss a fair coin three ties and define X = nuber of heads. HHH HHT HTH THH HTT THT TTH TTT X 3 0 P(X = 0) = P(X = ) = 3/8 P(X = ) = 3/8 P(X = 3) = x p(x) 0 3/8 3/8 3 Probability Histogra for x Probability Distributions The Mean and Standard Deviation Let X be a discrete rando variable with probability distribution p(x). Then the ean, variance and standard deviation of X are given as Mean : µ = x i p(x i ) i Variance : σ = (x i µ) p(x i ) Standard deviation : σ = σ i Toss a fair coin 3 ties and record X the nuber of heads. Find the ean and standard deviation of X. The probability distribution for X the nuber of heads in tossing 3 fair coins. Shape? Outliers? Center? Spread? Syetric; uniodal None µ =.5 σ =.688 5

6 3. Conditional probability 4. Independent and dependent events 5. P( B) =? 6. P( B) =? 7. Law of Total Probability 8. Bayes Rule 6

What is Probability? (again)

What is Probability? (again) INRODUCTION TO ROBBILITY Basic Concepts and Definitions n experient is any process that generates well-defined outcoes. Experient: Record an age Experient: Toss a die Experient: Record an opinion yes,