What Is the Probability of a Third Head After Two Heads with a Biased Coin?

[davidmoore63@y]

A random number p such that 0<p<1 is selected at random from a uniform distribution U[0,1]. A biased coin is then constructed such that the probability of heads on a single flip is p (thus 1-p for a tails).

This coin is flipped twice and the result is HH. If the coin is flipped a third time, what is the probability of a third head? More precisely, what is the fair value of a lottery ticket that pays one dollar if the third flip is a head, and zero otherwise? What would you pay for it/ sell it for?

 

[DEvens]

Is this a trick question?

"A biased coin is then constructed such that the probability of heads on a single flip is p..."

 

[davidmoore63@y]

not at all! I think it's well defined isn't it?

 

[DEvens]

Then the two H results don't give you any additional information. You know the probability is p, and you know the expected value of a $1 bet is p x $1.

 

[davidmoore63@y]

The question is asking for the probability of a third head PRIOR to finding out what p is.

 

[BWV]

the bet is worth $0 as the probability of drawing a rational number from the uniform distribution is zero and you can't pay someone and irrational number's worth of currency ;)

 

[davidmoore63@y]

Ok for you we round p to the nearest 1/100. You still have to do the question now!

 

[gill1109]

We want to know the probability of HHH given HH. It's Prob(HHH) / Prob(HH). The probability of HHH is the expectation value of p^3 where p is uniformly distributed on the interval [0, 1]. So it's int_0^1 p^3 dp = 1/4. Similarly Prob(HH) = int_0^1 p^2 dp = 1/3. So the answer is 3/4.

 

[davidmoore63@y]

Looks right to me