Slot Machine payout?

[SheetWise]

I was reading the post "Entering a Contest", and it made me think of problems created by playing against yourself. 30 years ago (pre PC era) I argued that the hold on a five line slot machine (three lines plus diagonals) would exceed the hold of the same number of coins played on a single line (assuming no 5th coin bonus).

My reasoning was that the reel symbols were staggered in such a way that a single win would nearly ensure there was no win on the other four lines, which was not true if the coins were played independently. My opponent argued that all lines were random selections, and how they were grouped made no difference. The obvious grouping was the jackpot symbol, which only appeared once on each reel.

I've never gone back to figure it out. Anyone know who was right?

[pzhon]

It looks like your friend was right.

E(A+B) = E(A) + E(B), even if A and B are not independent.

If the expected payout was the same on each line, there is no expected advantage or disadvantage to playing 5 lines at a time, even if the results of the lines are not independent.

[SheetWise]

This question has usually come up late at night, after a couple pitchers of beer. I think I need to run it. Most people have agreed with you, yet I still find it very counter-intuitive.

If my probability of hitting a jackpot remains constant -- then after hitting a jackpot with a single coin, my probability of hitting on the next pull is the same. Yet, if I hit the jackpot on one of five lines -- the jackpot was eliminated for the other four chances.

/images/graemlins/confused.gif

[pzhon]

[ QUOTE ]
if I hit the jackpot on one of five lines -- the jackpot was eliminated for the other four chances.

[/ QUOTE ]
Yes, and in the far more likely case that you miss the jackpot on the first line, the jackpot is slightly more likely for the other lines. These effects precisely balance out.

If you want to try to analyze an analogous situation, imagine you are playing roulette, and bet on one number at a time or bet on two consecutive numbers on the wheel. If the first number hits, the second definitely misses. If the first number misses, the second number is slightly more likely to hit. These effects precisely cancel out.

[bobman0330]

Hey, pzhon, i'm going to put up a rigorous demonstration of this point:
Imagine there's just the jackpot. 1 spot on the reel is the jackpot, and there are 9 blank spots. So, you win if the three reels end up with a jackpot straight across, or a jackpot running down the diagonal or up. Number the reel positions such that the top visible spot is 1, the middle spot 2, the bottom spot 3, and a 4-10 are not visible. You win when the three reels are 111, 222, 333, 123, or 321 (playing 5 coins).

If you play 1 coin, you win 1 out of a thousand times. ([1/10]^3).

If you play 5 coins:
If reel 1 is in position 1, you win when:
reel 2 is in position 2 and reel 3 is in 3 OR reel 2 is in 1 and reel 3 is in 1.
If reel 1 is in position 2, you win when:
reels 2 and 3 are in position 2.
If reel 1 is in position 3, you win when:
reel 2 is in position 2 and reel 3 is in 1 OR when reel 2 is in 3 and 3 is in 3.

SO:
P(winning, 5 coins)= 1/10*(1/100+1/100) + 1/10(1/100) + 1/10 (1/100+ 1/100)= 2/1000+1/1000+2/1000 = 5/1000 = 5 * P(winning, 1 coin)