# How much should one pay, answer using expectation of random variable?

In a gambling game, a woman is paid dollar 3 if she draws a jack or a queen and dollar 5 if she draws a king or an ace from an ordinary deck of 52 playing cards. If she draws any other card, she loses. How much should she pay to play if the game is fair?

• Do you know any formula for expected value ? Sep 16, 2015 at 9:01
• yes i do know the formula, what i need to know is what is the support of random variable in this question ? Sep 16, 2015 at 9:08
• i can use the formula but i need to understand where to start and how to start in these kind of problems ? Sep 16, 2015 at 9:10
• How much does she lose if she draw a card other than J,Q or K? Feb 4, 2017 at 3:59

To put you on track:

Let's say that she pays $l$ dollars to play.

Three things can happen and all with a certain probability: she wins $3$, she wins $5$ or she looses. Denoting the corresponding probabilities with $p_3$, $p_5$ and $p_l$ there is an expectation of: $$3p_3+5p_5-lp_l$$ A fair game means that this expectation equals $0$. Now start finding $l$.

• Nice approach there. Feb 4, 2017 at 4:00

Suppose C is the dollar cost of a fair game then expected gain/loss must be zero for a fair game.

P(win $\$3) = \dfrac8{52}\;\;$, P(win$\$5) = \dfrac8{52}$, thus

$E[X] = C - \left(3\cdot\dfrac{8}{52} + 5\cdot\dfrac{8}{52}\right) = 0$

Solving, we get $C = \dfrac{24 +40}{52} = \$1.23$• Thanks, could you please expalin what are the values taken by random variable in this question. how you defined random variable ? Sep 16, 2015 at 10:23 • The r.v. is the dollar value of the possible wins/losses on a turn. X took the values$3, 8$and$-C$dollars Sep 16, 2015 at 11:19 • User on ##statistics IRC found an error in this answer. According to text of the question, she is always paying for game, irrespective of win or loss (if she lose, she just paid for game and didn't win anything). Thus, the equation should be:$0 = \left(3 \times \frac{8}{52}\right) + \left(5 \times \frac{8}{52}\right) - C$The result is 1.23, which is what @Balraj wrote in a bit more confusing way. Please, extend the answer to explain what individual elements represent and correct the error Feb 3, 2017 at 22:37 • @Colombo: Oh, thanks, I mistook$3$and$5$as gain for the player. I will amend ! Feb 4, 2017 at 3:42 Finally i solved this problem , and thought it would be helpful for someone who needs a well explained answer , so here it goes..... Understand the language of question: "How much should she pay to play if the game is fair?" This simply means that on the average there should be no profit no loss in playing the game , in other words if the lady plays the game n times (n is very large no ) her money spent on the game should be equal to 0. Solving the problem: let z = " money paid by the lady " and let X is a random variable , which is the total money spent on one game. so now support of X (all the values that x can take) = (z-3), (z-5), z (z-3) when she draws jack or queen (z-5) when she draws ace or king z when she draws some other card now find probability of all these values of X P(z-3) = 2/13 , since there are 8 cards for jack and queen which gives probability = 8/52 = 2/13 p(z-5)= 2/13 since there are 8 cards for ace and king which gives probability = 8/52 = 2/13 p(z) = 9/13 since there is 36 other cards than ace, king, queen and jack now we find expected value of X and set it equal to 0 to find value of z E(X) = (z-3)*2/13 + (z-5)*2/13 + z*9/13 = 0  2z-6 +2z -10 + 9z = 0 13z= 16 z=1.23  This is kinda late, but hopefully this helps other people: 1. Let the discrete random variable$X$denote the outcome of a single game, where we define "outcome" to be either the receipt of money or the loss of money. 2. This discrete random variable has an associated probability distribution function$f(x)$. We can find the values of$f(x)$using standard counting methods/probability. Let us define$p$to be the amount of money you spent to play the game. Then there are three possible outcomes:$-p:$you won nothing, and you lost the amount of money you paid$3-p:$you gained this much money because you drew either a jack or a queen$5-p:$you gained this much money because you drew either a king or an ace We also define "fair" to be a game whose average result is$\geq 0$. In other words, on average, a player either doesn't lose money. Okay, on to the probabilities:$f(-p) = \frac{{36 \choose{1}}}{ {52 \choose 1}} = \frac{36}{52}$Repeat this process for the others, and you get$\frac{8}{52}$for each. All that remains is applying what you know about expected (average) values, setting the expected value expression to be greater than or equal to$0$, and solving for an appropriate$p$. The game will only be fair if the amount$p\$ that they're asking you to pay is less than or equal to some value.