Question I asked this question earlier today but did not get an answer. Thought I may have not sent it correctly the first time.
Our Civic Club is having a fundraiser. We have a Hi Lo Dice game. If the total count of a roll of 2 dice is 6 or below Lo wins. 8 or above Hi wins and 7 the house wins. A winner is paid $1 for each $1 bet, 1:1. If 8 people play and bet $1 per roll at random on either Hi or Lo for 30 rolls how much will the club make on an average?
If the club offers to pay 1:1 on Hi & Lo wins and 2:1 on a line bet/7 how much will they average on 30 rolls for 8 people?
If the club offers to pay 1:1 on Hi & Lo wins and 4:1 on a line bet/7 how much will they average on 30 rolls for 8 people?
Answer ed -
i can perhaps give you a push in the right direction to help you with your question.
the chance of a 7 is 6/36 = 1/6. then - by symmetry - the chance of lo = below 7 is the
same as that of hi = above 7 - so the chance of either is half of the remaining 30/36
or 15/36 = 5/12 each.
someone betting on hi will win 5/12 of the time and lose 5/12 + 1/6 = 7/12 of the time,
since they lose if either lo or 7 comes up. so on a $1 bet, they expect to lose
$7/12 - $5/12 = $2/12 = $1/6.
[if you prefer, imagine they play 12 times. they will expect to win $1 5 times and lose $1
7 times for a net loss of $2 in 12 plays - which averages to losing $2/12 = $1/6 per play.]
same goes for betting on lo.
so 8 players collectively expect to lose $1/6 x 8 = $8/6 = $4/3 = $1.33 per play.
multiply by 30 and you get that they collectively expect to lose $4/3 x 30 = $40 in 30 plays.
their loss is the house's gain - so the house expects to win $40 in 30 plays.
i don't quite understand the remaining parts of your question. i'm not sure what is meant by
"a line bet/7" and i'm not sure if "they" refers to the bettors or the house.
i can think of two meanings for "a line bet/7". to start, i'll suppose it means one can bet on
hi or lo or 7. one's expected winnings/play then depends on how often one bets on either hi or lo, vs how often on 7.
so - if one bets on 7 - with a $2 payoff - one expects to win $2 1/6 of the time and lose $1 5/6 of the time. so in 6 plays, say, one expects to win $2 on 1 play and lose $5 on the other 5 plays, for a net expected loss of $3 in 6 plays - or an expected loss of $1/2 per play. that is worse than betting on hi or low, where one expects to lose only $1/6 per play. so it actually makes no sense for anyone to bet on 7 with that betting scheme and payoff combo.
clearly the house does better if more people bet on 7 than on hi or lo.
if the payoff is $4 when betting on 7, the expected loss per play becomes $1/6 - the same as betting on hi or lo. in this case, the house does the same as when bettors can bet only on hi
or low.
but perhaps "a line bet/7" means one bets on hi or lo and in addition, can place a $1 side bet
on 7? in that case, adding up the expected losses for the hi/lo bet and the 7 bet, one expects
to lose $1/6 on the hi/lo bet and lose $1/2 more on the 7 side bet - so one expects to lose
$1/6 + $1/2 = $2/3 per play.
you can see from this that each dollar bet - on hi/lo or as a side bet - incurs its own expected gain or loss - and one just adds these up to get the total expected gain or loss for the entire
amount of dollars bet.
one can also get to this last result as follows: on 12 plays - betting on hi, say - one expects 'hi' 5 times for a gain of $0 [win $1 on hi, lose the $1 side bet], 'lo' 5 times - for a loss of $10 - and '7' twice - winning $1 each time [[win $2 on the side bet, lose $1 on hi/lo]. altogether, one expects to lose $8 in the 12 plays, for an average loss of $2/3 per play. whichever way you look at it, the house expects to gain $2/3 per player on each play.
what the house actually expects to gain altogether depends on the betting rules. must the hi/lo bet and the side bet be for the same amount? if one can bet different amounts on hi/lo and on the side bet, the player expects to lose $1/6 for each $1 bet on hi/lo and $1/2 for each $1 of the side bet. so the total amount the house expects to win per play from a player depends on how much the player bets each time on hi/lo and on the side bet.
similarly, with a $4 payoff for the 7 side bet - if the player bets $1 on hi/lo and $1 on a 7 side bet - the house expects to win $1/6 + $1/6 = $1/3 per play. in this case, even if one can bet different amounts on the two bets, what the house expects to win per play from the player is 1/6 of the total dollars the player bets. so the expected winnings to the house per play is just 1/6 of the total dollars all the players bet.
similarly, with a $2 payoff for a side bet on 7, the house expects to gain 1/6 of the total dollars bet on hi/lo plus 1/2 of the total dollars bet on 7.
