Question

A bag contains 3 gold marbles, 7 silver marbles, and 27 black marbles. Someone offers to play this game: You randomly select one marble from the bag. If it is gold, you win $3. If it is silver, you win $2. If it is black, you lose $1.

What is your expected value if you play this game?

Answer 1

The formula for Expected value is \(E\left(x\right)=\sum_{}^{ }n_{i}p_{i}\)

Essentially, you must gather all of the probabilities for each event, n. There is an event of being awarded $3, then $2, and then losing $1. They are all \(n_{i}\)

Now, you must gather the probability, p, for each event. For being awarded $3, it would be the probability of getting a gold marble. This would be 3 gold marbles over the total, 37.

Similar logic, for silver it is 7/37, and for black it is 27/37.

Now you have to plug in. Remember that the ∑ just means to add every \(n_ip_i\) you get.

\(E\left(x\right)=3\left(\frac{3}{37}\right)+2\left(\frac{7}{37}\right)-1\left(\frac{27}{37}\right)\)

You can evaluate from here.