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## Main Question or Discussion Point

Hi all,

I am creating a game for fun, which need some math skill to work out the chance of winning and the way to keep the banker never lose. The configuration of the game is like this: five boxes marked no.1, no.2, no.3, no.4 and no.5; there are many balls in different color in each box. For each game cost 1 token to play. Player drawing one ball from each box from no.1 to no.5. If player get 5 red balls, he/she will be given 1250 tokens as reward.

The chance to get 1 red ball from no.1 box is 10%

The chance to get 1 red ball from no.2 box is 80%

The chance to get 1 red ball from no.3 box is 50%

The chance to get 1 red ball from no.4 box is 30%

The chance to get 1 red ball from no.5 box is 20%

Based on my understanding from stat class, the expectation of winning this game is

##0.1*0.8*0.5*0.3*0.2*1250 \text{(tokens)} = 3 \text{tokens}##

This means on average players will win 3 tokens back. The banker add a new rule to the game that if player get 4 red balls from no.2, no.3, no.4 and no.5 box, he/she could draw one ball once from the no.1 box again (with the drawn ball put back to the box) by paying extra tokens. This sounds good chance for player since three is 10% of chance to win 1250 tokens in this case. However, in order not to lose, banker must increase the extra token for redraw. I am thinking how many tokens should be asked for. My first thought, since the expectation of winning 5 red balls in 3 (tokens), each redraw should cost 3 tokens to compensate that. I did the simulation, running the game for 10000000 times with proper drawing probability for red ball in each box. It turns out the expectation value is very close to 3 (without redraw); however, with redraw case, if I ask 3 tokens as extra cost, it gives expectation of 5.42 instead of 3. So asking 3 tokens for redraw is not correct to balance the lose. There must be something I don't understand correctly from my math. Any one could give me some hints on that. Thanks

I am creating a game for fun, which need some math skill to work out the chance of winning and the way to keep the banker never lose. The configuration of the game is like this: five boxes marked no.1, no.2, no.3, no.4 and no.5; there are many balls in different color in each box. For each game cost 1 token to play. Player drawing one ball from each box from no.1 to no.5. If player get 5 red balls, he/she will be given 1250 tokens as reward.

The chance to get 1 red ball from no.1 box is 10%

The chance to get 1 red ball from no.2 box is 80%

The chance to get 1 red ball from no.3 box is 50%

The chance to get 1 red ball from no.4 box is 30%

The chance to get 1 red ball from no.5 box is 20%

Based on my understanding from stat class, the expectation of winning this game is

##0.1*0.8*0.5*0.3*0.2*1250 \text{(tokens)} = 3 \text{tokens}##

This means on average players will win 3 tokens back. The banker add a new rule to the game that if player get 4 red balls from no.2, no.3, no.4 and no.5 box, he/she could draw one ball once from the no.1 box again (with the drawn ball put back to the box) by paying extra tokens. This sounds good chance for player since three is 10% of chance to win 1250 tokens in this case. However, in order not to lose, banker must increase the extra token for redraw. I am thinking how many tokens should be asked for. My first thought, since the expectation of winning 5 red balls in 3 (tokens), each redraw should cost 3 tokens to compensate that. I did the simulation, running the game for 10000000 times with proper drawing probability for red ball in each box. It turns out the expectation value is very close to 3 (without redraw); however, with redraw case, if I ask 3 tokens as extra cost, it gives expectation of 5.42 instead of 3. So asking 3 tokens for redraw is not correct to balance the lose. There must be something I don't understand correctly from my math. Any one could give me some hints on that. Thanks