7.7 Jane Street Phone Interview

[alexanderw]

X: # Heads, X~BIN(5,1/2), E(X)=5*1/2=5/2=2.5, x = 0,1,..,5. We know the expectation of a discrete distribution is calculated by summing up the probabilities over all possible x. But here in this specific setting P(HHHHH) is not possible and P(HHHHT) is also not possible so they must be subtracted. But you also have to take into account P(HHHH).

So the expected value of this game is 2.5 - 5*(1/2)^5 - 4*(1/2)^5 + 4*(1/2)^4 = 2.5 - 1/32 = 2.46875

 

[Spen]

As I see, there are four coins, and the expected payoff in first round is $2 as all said. You get to flip a second time only one coin only if all four didn't show up H in the first. And the probability of getting a chance for a second flip is 1-(1/2)^4=15/16 i.e. a little less than 1.
Irrespective of which coin you chose for second time, the payoff would be (15/16) * 0.5 making a total of

2 + [ (15/16)*0.5] = $2.46874

Why is the last part *0.5 but not *1 though?