Question

Suppose you want to play a carnival game that costs 7 dollars each time you play. If you win, you get $100. The probability of winning is 1⁄50. What is the expected value of the amount that you, the player, stand to gain?

Answer 1

Answer:

The correct answer would be $5

Explanation:

The formula to use is "Expected return to player" which is

E(x) = x.p(x)

where x is the return to player if they win

and p(x) is the probability of winning.

So here,

x = $100 (return to player for winning)

p(x) = 1/50 (probability of winning)

Therefore expected return to player is

E(x) = x.p(x)

= $100 x 1/50

= $100/50

= $2

Cost: $7

Expected return to player is $2.

Therefore Loss (to player) is Cost minus Expected return

= $7 - $2 = $5 <---- expected value for the carnival to gain,

The loss to the player is the carnival's gain. It's $5.

Answer 2

There is a loss of $\bf \ \ 5$ to the player in carnival game.

Further explanation:

In the question it is given that in a carnival game it costs $\ \ 7$ to play each time.

The amount of money received after winning a game is $\ \ 100$.

The probability of winning a game is $\frac{1}{50}$.

Consider the probabilityof winning a game as $P$.

The probablity of lossing a game is calculated as follows:

$\begin{aligned}P'&=1-P\\&=1-\dfrac{1}{50}\\&=1-0.02\\&=0.98\end{aligned}$

Consider the amount of money received by winning a game as $X$.

Expected return to player is calculated as follows:

$\boxed{\text{E}(X)=XP(X)}$                          ......(1)

If a person won a game then he must received $\ \ 100$ and the amount of money paid for playing one game is  $\ \ 7$.

This implies that in winning a game there is a gain of $\ \ 93$ and if a player lost a game then there is a loss of $\ \ 7$.

Figure 1 (attached in the end) represents the different cases of winning and loosing.

Using the equation (1), the expected value of the amount to be received by a player is calculated as follows:

$\begin{aligned}\text{E}(x)&=\left(93\cdot 0.02\right)+\left((-7)\cdot 0.98\right)\\&=1.86-6.86\\&=-5\end{aligned}$

The value of $\text{E}(x)$ as obtained above is $-5$ which is a negative value.

The negative value $\text{E}(x)$ implies that there is a loss to the player of $\ \ 5$.

Thus, there is a loss of $\bf \ \ 5$ to the player in carnival game.

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Answer details:

Grade: High school

Subject: Mathematics

Chapter: Probability

Keywords: Probabilty, expected value, loss, profit, carnival game, 7 dollars, win $100, 1/50, winning, expected return, mathematics, sample space, money gain, money paid.