3. A fair coin is flipped, and a wheel is spun. The wheel is balanced and takes

Question

3. A fair coin is flipped, and a wheel is spun. The wheel is balanced and takes values in x; (0, 1); the spin is represented by a . If the flip resulted in "heads", we observe the value y if the flip resulted in "tails" we observe the value y2n (a) Find the probability that y is greater than 1/2 if heads resulted (b) Find the probability that y is greater than 1/2 if tails resulted. (c) Find the probability that y is greater than 1/2. (d) Find the probability that heads resulted if y is greater than 2/3. (e) Find the probability that tails resulted if y is greater than 2/3.

Explanation / Answer

as for uniform distribution F(x) =P(X<x) =x for 0<x<1

a)P(y>1/2|heads) =P(X>1/2) =1-P(X<1/2) =1-1/2 =1/2

b)P(y>1/2|tails) =P(2x>1/2) =P(x>1/4) =1-P(X<1/4) =1-(1/4) =3/4

c)

P(y>1/2) =P(heads)*P(y>1/2|heads)+P(tails)*P(y>1/2|tails) =(1/2)*(1/2)+(1/2)*(3/4)=5/8

d)

P(y>2/3) =P(heads)*P(y>2/3|heads)+P(tails)*P(y>2/3|tails) =(1/2)*P(X>2/3)+(1/2)*(x>1/3)

=(1/2)*(1-2/3)+(1/2)*(1-1/3) =(1/2)*(1/3)+(1/2)*(2/3) =1/2

hence P(heads|y>2/3) =P(heads)*P(y>2/3|heads)/P(y>2/3) =(1/2)*(1/3)/(1/2 )=1/3

e)P(tails|y>2/3) =1-P(heads|y>2/3) =1-1/3 =2/3

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