I heard this puzzle a while back and wasn't sure how to answer it:
You play a game whose cost to play goes up at every round, and there are 2 possible outcomes. Winning and losing. You win with 60% likelyhood and lose with 40% chance at all rounds.
Round 1 - Costs \$1 to play On the first round, if you win, you win another \$1. If you lose, you lose your \$1.
Round 2 - Cost \$2 to play On the second round, if you win, you win another \$1. If you lose, you lose your \$2.
Round 3 - Cost \$3 to play On the Thirdround, if you win, you win another \$1. If you lose, you lose your \$3.
What are the expected values at each round, and how do you calculate them?
Cost \$2 and you win another \$1 means that if you win you cash \$3?, Right? This because usually the cost of a game is never reimbursed. Have I understood well? And you want to know the expected gain at EACH round? This independently about what happened before?
Under these assumptions, the three expectations are the following:
$$\mathbb{E}[R_1]=1\times0.6-1\times0.4$$
$$\mathbb{E}[R_2]=1\times0.6-2\times0.4$$
$$\mathbb{E}[R_3]=1\times0.6-3\times0.4$$