Question

You are trying to decide whether to play a carnival game that costs $1.50 to play. The game consists of rolling 3 fair dice. If the number 1 comes up at all, you get your money back, and get $1 for each time it comes up. So for example if it came up twice, your profit would be $2. The table below gives the probability of each value for the random variable X, where:

X = profit from playing the carnival game (in $).

What happened?	X = profit in $	P(X)
Did not roll any 1's	-$1.50	0.5787
Rolled a single 1	$1.00	0.3472
Rolled two 1's	$2.00	0.0694
Rolled three 1's	$3.00	0.0046

a. Find the expected value of the random variable X. Keep in mind it may be negative. Round your answer to 2 decimal places.

b. If you played the carnival game 1000 times, how much money would you expect to lose? Round your answer to the nearest whole number.

Interpret your findings. Select your answer from one of the following options.

• a.The expected value is small but negative, so in the long term, if you play the game a large number of times, even though you lose a little bit, when you win it will balance it out, and you will break even.
• b.The expected value is small but negative, so in the long term, if you play the game a large number of times, even though you will lose a little bit some times, you will win more times so you will have more money than when you started.
• c.The expected value is small but negative, so in the long term, if you play the game a large number of times, even though you will win some, you will have less money than when you started.

Answer 1

Question (a)

Expected value = X_i * P(X_i)

So expected value of the random variable X here = (-1.5) * 0.5787 + 1.00 * 0.3472 + 2.00 * 0.0694 + 3.00 * 0.0046

= -0.8681 + 0.3472 + 0.1388 + 0.0138

= -0.36825

= -$0.37

Question (b)

If you played the carnival game 1000 times, how much money would you expect to lose

Money you expect to lose = 1000 * (-0.36825)

= -368.25

= -$368

So you expect to lose $368

Question (c)

Thoguh you play a large number of games, at the end you will always loose money because the expected value of the game here is negative

The expected value is small but negative, so in the long term, if you play the game a large number of times, even though you will win some, you will have less money than when you started.

So Answer is Option C