First we need to define the betting methods.
METHOD 1: You bet an amount $W, the wager, to win a prize with a value of $V. If you win, you take the prize and your wager is refunded, so your net profit is $V. If you lose, your wager is not refunded, so your net profit is -$W (a loss of $W).
METHOD 2: This is like Method 1, but whether you win or lose, your wager is not refunded. Your net profit is $V-$W if you win and -$W if you lose.
Now we come to the game. Imagine a room with 8 tables, and each table has a gambler and a banker who takes the bet; and in the centre of the room is a spinning wheel divided into 8 equal segments. One of the segments is the winning segment, but it is not marked. Only when the wheel stops will it be randomly determined by a machine which segment is the winner.
The 8 tables represent the 8 possible outcomes when the wheel is spun, and the gamblers place their bets. Each table wagers $W for a different segment, so table 1 bets on segment 1, table 2 on segment 2, and so on. Only one table can win and the probability of winning is ⅛, or, in other words, the odds are 8:1 against.
The wheel is spun. There will be one winning table and 7 losers.
If table 1 wins (segment 1 is the winner), the gambler under Method 1 wins $V. The net profit is $V, and we’re told that the profit is 4 times the wager, so V=4W. Under Method 2, the net profit is V-W=4W, so V=5W.
The average profit over all the tables under Method 1 is (4W-7W)/8=-3W/8.
The average profit under Method 2 is (5W-W-7W)/8=-3W/8.
So, it doesn’t matter which betting method is used, the average profit (expected value)=-⅜W dollars (loss of ⅜W dollars).
Note, however, that the prize value is related to the wager, so if, for example, regardless of the betting method, $8000 is the prize, the wager would be $2000 under Method 1; but it would be $1600 under Method 2, so the actual value of the loss would be different: $750 loss for Method 1 and $600 loss for Method 2.
