The Powerball lottery jackpot has a winning probability of 1/189,000,000. If you only consider the options of winning the jackpot ($100,000,000) or losing, what the is the mean value of a ticket that costs $2 to buy? At what point does the mean value of a ticket become positive?

In this case, we can write the mean value as:

MV = ∑pₙxₙ - $2

Where pₙ is the probability of event xₙ

in this case we consider only two events:

x₁ = winning $100,000,000, p₁ = 1/189,000,000

x₂ = not winning (or winning $0.0), p₂ = 1 - p₁.

Then the mean value of the ticket is:

MV = $100,000,000*1/189,000,000 + 0*(1 - 1/189,000,000) - $2 = $0.53 - $2

MV = -$1.47

b) At what point the mean value becomes positive?

Well, if the ticket costs less than $0.53, the value would become positive.

if we also consider other possible prices, we will have other positive components in the sum above, then the mean value may become positive.

Now, you may notices that it does not matter if you buy N tickets, the mean value will always be negative becuase:

MV = N*$0.56 - N*$2 = N*-$1.47

This is because yes, as more tickets you buy, the more probabilities of winning the prize you have, but if you buy N tickets, you need to pay N times $2, so the mean value actually decreases.