Question

Suppose that the probability, X, that a randomly selected coin from a large collection comes up heads when tossed has a uniform distribution on [0,1].

a) Determine the probability of the event, HH, that a coin selected at random from the collection comes up heads both times when tossed twice.

b) Determine the conditional p.d.f of X given the event HH.

c) Determine the probability the same coin comes up heads again if it tossed a third time.

Answer 1

Suppose that the probability, X, that a randomly selected coin from a large collection comes up heads when tossed has a uniform distribution on [0,1] that means if headcomes with probability 0.5.

a) the probability of the event, HH, that a coin selected at random from the collection comes up heads both times when tossed twice is

x1=runif(100,0,1)
x2=runif(100,0,1)
p1=length(x1[x1<=0.5])/100
p1
p2=length(x2[x2<=0.5])/100
p2
p=p1*p2
p

0.2448

b) the conditional p.d.f of X given the event HH

c)

> x1=runif(100,0,1)
> x2=runif(100,0,1)
> x3=runif(100,0,1)
> p1=length(x1[x1<=0.5])/100
> p1
[1] 0.52
> p2=length(x2[x2<=0.5])/100
> p2
[1] 0.53
> p3=length(x3[x3<=0.5])/100
> p3
[1] 0.49
> p=p1*p2*p3
> p
[1] 0.135044
>