Suppose I have a box where you can pick to receive 50 or a second box. If you pick 50 you walk away with 50 but if you pick the second box you then will have a 50% chance of receiving 1 or 50% chance of receiving another box. This process keeps repeating up to 4 times, see below.
Start off box 0 Box 0 contains 50 or Box 1, you can pick Box 1 contains 50% chance at 1 or 50% chance at Box 2 Box 2 contains 105 or Box 3, you can pick Box 3 contains 50% chance at 1 or 50% chance at Box 4 Box 4 contains 221 or Box 5, you can pick Box 5 contains 50% chance at 1 or 50% chance at Box 6 Box 6 contains 465 or Box 7, you can pick Box 7 contains 50% chance at 1 or 50% chance at Box 8 Box 8 contains 977, this is the last box
How would you play this game to maximise the reward? Based on my calculations it seems like it's always worth it to go for box 8 and 977 since the expected value is 62?
My calculation is as follows.
Expected value = 0.51 + 0.251 + 0.1251 + 0.06251 + 0.0625*977 = 62, this is greater than the initial choice of picking 50 so you should always go until the last box.