In Las Vegas the roulette wheel has a 0 and a 00 and then the numbers 1 to 36 marked on equal slots; the wheel is spun and a ball stops randomly in one slot. When a player bets 1 dollar on a number, he receives 36 dollars if the ball stops on this number, for a net gain of 35 dollars; otherwise, he loses his dollar bet. Find the expected value for his winnings.
To find expected value, we use the formula:
\(E(x) = ∑x^ip(x^i)\)
where \(x^i\) is the outcome and \(p(x^i)\) is probability of the outcome.
The probability of winning would be \(\frac{1}{38}\) since there are 38 number slots (0,00,1-36). This would be multiplied by the outcome which would be the net gain of 35 dollars.
The probability of losing would be \(\frac{37}{38}\) since there are 38 number slots (0,00,1-36). This would be multiplied by the outcome which would be the minus 1 dollar for playing.
So putting both together, the expected value is :
35(\(\frac{1}{38}\))+-1(\(\frac{37}{38}\)) –> 0.9210526 -
0.9736842 = -0.05263157
In a second version of roulette in Las Vegas, a player bets on red or black. Half of the numbers from 1 to 36 are red, and half are black. If a player bets a dollar on black, and if the ball stops on a black number, he gets his dollar back and another dollar. If the ball stops on a red number or on 0 or 00 he loses his dollar. Find the expected winnings for this bet.
Using the same formula:
The probability of winning would be \(\frac{18}{38}\) since half the numbers of 1 to 36 being the chosen colors gives us 36/2 = 18. The 0 and 00 slot were not given a color in the description so you cannot win red or black on them. This would be multiplied by the outcome which would be a 1 (the bonus dollar).
The probability of losing would be \(\frac{20}{38}\) due to the 18 out of 36 number slots that are the opposite color of your bet and the 2 non-coloured slots (0 and 00) This would be multiplied by the outcome which would be the minus 1 dollar for playing.
So putting both together, the expected value is :
1(\(\frac{18}{38}\))+-1(\(\frac{20}{38}\)) –> 0.47368421 -
0.52631578 = -0.052631578