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Review the concept of expected value; E[X]_what exactly is it? Let X = the net gain to player in game of chance: If E(x) is positive, this gives the average amount ...

Question

Review the concept of expected value; E[X]_what exactly is it? Let X = the net gain to player in game of chance: If E(x) is positive, this gives the average amount theplayer can expect to win per play if helshe repeats the game over and over, If E(x) is negative, this give the average amount the player can expect t0 lose per play if helshe repeats the game over and over. (NOTE over several repeated games the player wlll win few; but lose on many):a) Complete the probability distribution for Ame

Review the concept of expected value; E[X]_what exactly is it? Let X = the net gain to player in game of chance: If E(x) is positive, this gives the average amount the player can expect to win per play if helshe repeats the game over and over, If E(x) is negative, this give the average amount the player can expect t0 lose per play if helshe repeats the game over and over. (NOTE over several repeated games the player wlll win few; but lose on many): a) Complete the probability distribution for American roulette: The American roulette wheel has 38 numbers" 00, 0, 1, 2 34, 35, 36 If a player bets $1 on one number and wins_ the player keeps the dollar and recelves an additional 35 dollars Otherwise , Ihe dollar I8 lost. (write each P(x) asan exact fractlon) Gain; P(x) What is the expected value of the player $ gainiloss? (Show the work by hand) E(x) =lr (x " P())



Answers

A roulette wheel has 38 slots. Thirty-six slots are numbered from 1 to $36 ;$ the remaining two slots are numbered 0 and 00 . Suppose the "number" 00 is considered not to be even, but the number 0 is still even. In a $\$ 1$ bet on even, the bettor pays $\$ 1$ to play. If the ball lands in an even numbered slot, he receives back the dollar he bet plus an additional dollar. If the ball does not land on an even numbered slot, he loses his dollar. Let $X$ denote the net gain to the bettor on one play of the game. a. Construct the probability distribution of $X$. b. Compute the expected value $E(x)$ of $X,$ and explain why this game is not offered in a casino (where 0 is not considered even). c. Compute the standard deviation of $X$.

Here on this problem we're talking about expected by the expected value is the theoretical average. And we're dealing in American religion in roulette that we have 38 numbers. If a player bets $1 on a number of wins, the player keeps the dollar and receives an additional $35. Otherwise, the dollar is lost. So are expected value. Here there is a one out of 38 chance the person receives $35. And so there's a 37 out of 38 chance that the person loses the dollar entirely. And so when we evaluate this 35/38 minus 37/38 this gives us negative 0.5 And so on average, the person is expected to lose five cents.

For this problem we are asked to calculate the expected net gain for the player for one play of the game of roulette. We're told that in roulette the wheel has 38 numbers 00012, up to 34 35 36 marked on equally spaced slots. If a player bets $1 on a number and wins, then the player keeps the dollar And receives an additional $35. Otherwise the dollar is lost. So what we can do to calculate the expected value here is recognize that We have, if we win we get $35, so we have a value of 35 and then the probability of winning is going to be one in 38. You have to get the right number. Then the other possible outcome is that we lose a dollar And the probability of that happening is 37 in 38. So adding that together and calculating it out, The net are the expected value of the expect expected net gain is going to be negative 1/19 or in decimal. That's going to be approximately negative 0.5. So you'd expect to lose five cents

Okay so we're just gonna verify the probabilities add up to one first part. So that's .474 Plus .5- six. And that indeed does equal one sort of probability thing is verified. So for part B. Or asked to see what is the expected value of the random variable X. So the expected value is going to be the mean mean is going to be calculated as -1 times 474 plus. Well this one's negative one this one's positive plus a negative one times 5 to 6. Mhm mm. And this is going to be equal to sure .474 times -1-6. This is going to be equal to negative point oh 52 She expected to lose about five cents. Yeah same thing for C. If I can find it On average how much will use for play. Well that's exactly how much you're gonna be losing. Its gonna be here five cents. About 5.2 cents. And for party personally, how much would you expect to lose if you bet a dollar on red 100 times? Okay, so that's just gonna be five to 5.2 cents that you lose times 100. It's just gonna equal $5.20. And if you bet 1000 times you would lose $52 for 1000. And is our part e there is a roulette profitable game to play. You guys probably already know the answer. That. Mm.

So we have here is a relent wheel. Um And for those of you who aren't familiar with the roulette wheel, we have a bunch of different spaces that can be landed on, And there are 38 slots and they are numbered 000 12, and that goes all the way to 36. So we have a ball that spun around the wheel and it has to fall into a slot. Um So we want to construct a probability distribution for the winnings on each spin. So for a what we have is X p X just like this, And then we have 35 And p of X can be 0.026, three and negative one 0.97 36. So that's the probability that we lose the dollar, is the probability that we make The $35. Then we want to see for part B. Mhm, determining the mean and standard deviation. So remember mean, Is this some of all values divided by the number of values, So for part B We end up getting negative 0.05, and for the standard deviation, that's going to be equal to 576 Then for part C, were asked um That we're told that we played a game 100 times, so we want to find the sampling distribution and that's going to be, and this right here is equal to this over The square root event, so it's gonna be 576 over the square root of 100. Then we want to know the probability of being ahead after playing the game 100 times. So for d If we play the game 100 times, we find that we only have a 46 .5 4 chance of being ahead. Um So even if we play multiple times, we're still going to be losing money, And we want to know the probability that the sample mean is greater than zero for an equals 100 next. We want to know the probability of being ahead in the game after 200 times. So for e we end up seeing that that's a lower percentage, 04511 and then we want to see 1000 times, and that's going to be 0.39 18. So we see that the lesson is to quit while you're ahead and gambling, because the more that you keep going trying to catch up again, the more you could end up just losing out.


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