Answer to Question #91983 in Statistics and Probability for joe

Question #91983
A number x is selected at random in the interval [-1, 2]. Let the events A={x<0}, B={|x-0.5|<0.5}, and C=x>0.75. Find P[A|B], P[B|C], P[A|C^c] and P[B|C^c].
1
Expert's answer
2019-07-29T09:45:46-0400

Assume that the probability of any subinterval I of [-1, 2] is proportional to its length


P(I)=klength(I)P(I)=k \cdot length(I)

If we let I=[2,1],I=[-2, 1], then we must have


1=P(S)=P([2,1])=klength([2,1])=k31=P(S)=P([-2, 1])=k\cdot length([-2,1])=k\cdot 3k=13k={1 \over 3}P[A]=13length([1,0))=131=13P[A]={1 \over 3}length([-1,0))={1 \over 3}\cdot 1={1 \over 3}

x0.5<0.5=>0.5<x0.5<0.5=>0<x<1|x-0.5|<0.5=>-0.5<x-0.5<0.5=>0<x<1


P[B]=13length((0,1))=131=13P[B]={1 \over 3}length((0,1))={1 \over 3}\cdot 1={1 \over 3}

P[C]=13length((0.75,2))=131.25=512P[C]={1 \over 3}length((0.75,2))={1 \over 3}\cdot 1.25={5 \over 12}

P[AB]=P[AB]P[B]=0P[A|B]={P[A\cap B] \over P[B]}=0

P[BC]=13length((0.75,1))=130.25=112P[B\cap C]={1 \over 3}length((0.75,1))={1 \over 3}\cdot 0.25={1 \over 12}

P[BC]=P[BC]P[C]=112512=15P[B|C]={P[B\cap C] \over P[C]}={{1 \over 12} \over {5 \over 12}}={1 \over 5}

P[CC]=1512=712P[C^C]=1-{5 \over 12}={7 \over 12}

P[ACC]=13length([1,0))=131=13P[A\cap C^C]={1 \over 3}length([-1,0))={1 \over 3}\cdot 1={1 \over 3}


P[ACC]=P[ACC]P[CC]=13712=47P[A|C^C]={P[A\cap C^C] \over P[C^C]}={{1 \over 3} \over {7 \over 12}}={4 \over 7}

P[BCC]=13length(0,0.75])=130.75=14P[B\cap C^C]={1 \over 3}length(0,0.75])={1 \over 3}\cdot 0.75={1 \over 4}

P[BCC]=P[BCC]P[CC]=14712=37P[B|C^C]={P[B\cap C^C] \over P[C^C]}={{1 \over 4} \over {7 \over 12}}={3 \over 7}


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