# Net Gain in Roulette Betting - Expectation & Standard Deviation

• habman_6
In summary, the roulette wheel has seven equally likely outcomes, with an even number outcome resulting in a $10 payout. If you bet on an odd number, your payoff is either$10 or -10, depending on whether or not the odd number comes up. After 400 turns, a player's expected net gain is -571 and their standard deviation is 198.
habman_6
30) A special roulette wheel has seven equally likely outcomes: 0, 1, 2, 3, 4, 5, 6. If you bet that an odd number comes up, you win or lose $10 according to whether or not that event occurs. If X denotes your ‘net’ gain, X=10 if we see 1, 3, or 5, and X = -10 otherwise. Suppose that you play this game 400 times. Let Y be your net gain after these 400 plays. The mean (expectation) and standard deviation of Y are, respectively (accurate to the number of figures shown): A) -270; 198 B) -571; 198 C) -270; 988 D) -571; 988 E) none of the other answers displayed I can get the mean (-571), but for the standard deviation, I only get the answer (apparently 198) when I use the formula: sigma² _ new = b * sigma²_old When the formula SHOULD use b², instead of just b. (Im using sigma² = Sum((x_i-u)²*p) to get the original variance.) habman_6 said: 30) A special roulette wheel has seven equally likely outcomes: 0, 1, 2, 3, 4, 5, 6. If you bet that an odd number comes up, you win or lose$10 according to whether or not that event occurs. If X denotes your ‘net’ gain, X=10 if we see 1, 3, or 5, and X = -10 otherwise. Suppose that you play this game 400 times. Let Y be your net gain after these 400 plays. The mean (expectation) and standard deviation of Y are, respectively (accurate to the number of figures shown):
A) -270; 198
B) -571; 198
C) -270; 988
D) -571; 988
E) none of the other answers displayed

I can get the mean (-571), but for the standard deviation, I only get the answer (apparently 198) when I use the formula:

sigma² _ new = b * sigma²_old

When the formula SHOULD use b², instead of just b.

(Im using sigma² = Sum((x_i-u)²*p) to get the original variance.)
On anyone spin your probability of winning is 3/7 and of losing is 4/7. This is a binomial distribution with p= 3/7, q= 4/7 and N= 400. The mean value, and standard deviation for a binomial distribution with p, q, n are np and $\sqrt{npq}$ respectively. With your values, yes, -571 is the expectation and 988, not 198, is the is the standard deviation. I can't say anything about your formula, sigma² _ new = b * sigma²_old, since you haven said what "b" and "sigma_old" are.

What I did was the mean is:

$$400*[(3/7)(10)+(4/7)(-10)]$$
= -571

Now, for standard deviation, it should be:

(For a single turn) variation =

$$sigma^2=[10-(-1.428)]^2*(3/7)+[-10-(-1.428)]^2*(4/7)$$

So, sigma_new² (after 400 turns) SHOULD =

b^2*sigma^2
=400²*([10-(-1.428)]²*(3/7)+[-10-(-1.428)]²*(4/7))²

However, I only get the right answer (which is 198), when I simply use 400 instead of 400².

Last edited:
Nevermind, I got it using binomial distribution. The formula I was using before was for when you linearly transform by multiplying by 400, not when you repeat 400 times.

Thanks

## 1. What is the concept of "Net Gain" in roulette betting?

"Net Gain" refers to the overall profit or loss that a player experiences after a series of roulette bets. It takes into account both the amount of money won and lost during the betting process.

## 2. How is "Net Gain" calculated in roulette betting?

To calculate "Net Gain", the total amount of money won is subtracted from the total amount of money lost. This will give the player an overall profit or loss amount.

## 3. What is the role of "Expectation" in roulette betting?

"Expectation" refers to the predicted average outcome of a roulette bet based on the probabilities of winning and losing. It can help players make informed decisions about their betting strategy.

## 4. How does "Standard Deviation" affect roulette betting?

"Standard Deviation" is a measure of the variability or spread of a set of data. In roulette betting, it can be used to assess the risk associated with a particular betting strategy. Higher standard deviation means a higher risk of losing, while lower standard deviation means a lower risk.

## 5. Is there a guaranteed way to achieve a positive "Net Gain" in roulette betting?

No, there is no guaranteed way to always achieve a positive "Net Gain" in roulette betting. The game is based on chance and there is always a risk of losing money. However, players can use strategies and manage their bets carefully to try and increase their chances of winning.

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