Question

question 4
4 pts

amita pays $0.88 to play a game at a local fair that involves receiving the number of quarters that corresponds to the number that comes up on the roll of a die. using the expected value of the game, compute the expected gain or loss for amita.

note: please avoid rounding numbers in the middle of your calculations. however, round your final answer to two decimal places, before entering it in the box below. a negative final answer indicates an expected loss for amita for playing the game.

Explanation:

Step1: Calculate the expected value of quarters

The probability of each outcome (1 - 6) when rolling a die is \(p=\frac{1}{6}\). The value of quarters for outcome \(i\) is \(0.25i\) (since 1 quarter = \(0.25\) dollars).
The expected value \(E(X)\) of a discrete - random variable \(X\) is given by \(E(X)=\sum_{i = 1}^{n}x_ip_i\). Here, \(n = 6\), \(x_i=0.25i\), and \(p_i=\frac{1}{6}\) for \(i = 1,2,\cdots,6\).

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Using the sum of an arithmetic series formula \(S_n=\frac{n(n + 1)}{2}\) with \(n = 6\), \(S_6=\frac{6\times(6 + 1)}{2}=21\).
So, \(E(X)=\frac{0.25\times21}{6}=\frac{5.25}{6}=0.875\)

Step2: Calculate the expected gain or loss

The cost to play the game is \(C = 0.88\) dollars.
The expected gain or loss \(G\) is \(G=E(X)-C\)
Substitute \(E(X)=0.875\) and \(C = 0.88\) into the formula:
\(G=0.875-0.88=- 0.005\)

Answer:

\(-0.01\)