A bag contains 2 gold marbles, 6 silver marbles, and 23 black marbles. Someone offers to play this game: You randomly select one marble from the bag. If it is gold, you win $4. If it is silver, you win $2. If it is black, you lose $1.

What is your expected value if you play this game?

Well, let's calculate your expected value and find out!

The probability of selecting a gold marble is 2/31, so you have a 2/31 chance of winning $4. The probability of selecting a silver marble is 6/31, so you have a 6/31 chance of winning $2. The probability of selecting a black marble is 23/31, so you have a 23/31 chance of losing $1.

Now let's do some math!

Expected value = (probability of winning * amount won) - (probability of losing * amount lost)

Expected value = [(2/31 * $4) + (6/31 * $2)] - (23/31 * $1)

Expected value = ($8/31 + $12/31) - ($23/31)

Simplifying further:

Expected value = ($20/31) - ($23/31)

Expected value = -$3/31

So, your expected value if you play this game is -$3/31. However, keep in mind that expected value is just a mathematical concept. It doesn't guarantee actual results, so play at your own risk!

To calculate the expected value, we need to determine the probability of each outcome and multiply it by the corresponding value.

1. Probability of selecting a gold marble:
There are 2 gold marbles out of a total of 2 + 6 + 23 = 31 marbles.
So, the probability of selecting a gold marble is 2/31.

2. Probability of selecting a silver marble:
There are 6 silver marbles out of the same total of 31 marbles.
Thus, the probability of selecting a silver marble is 6/31.

3. Probability of selecting a black marble:
There are 23 black marbles out of the same total of 31 marbles.
Therefore, the probability of selecting a black marble is 23/31.

Now let's calculate the expected value:

Expected value = (Probability of gold marble * Value of gold marble) + (Probability of silver marble * Value of silver marble) + (Probability of black marble * Value of black marble)

Expected value = (2/31 * 4) + (6/31 * 2) + (23/31 * -1)

Expected value = 8/31 - 12/31 - 23/31 = -27/31 ≈ -0.87

Therefore, the expected value of playing this game is approximately -$0.87.

To calculate the expected value, we need to multiply the value of each outcome by its probability and then sum them up.

Let's start by calculating the probability of drawing each type of marble:

- There are 2 gold marbles out of a total of 2 + 6 + 23 = 31 marbles. So, the probability of drawing a gold marble is 2/31.
- There are 6 silver marbles out of a total of 31 marbles. So, the probability of drawing a silver marble is 6/31.
- There are 23 black marbles out of a total of 31 marbles. So, the probability of drawing a black marble is 23/31.

Now let's calculate the expected value:

The expected value is given by the following formula:
Expected Value = (Probability of Gold Marble) * (Value of Gold Marble)
+ (Probability of Silver Marble) * (Value of Silver Marble)
+ (Probability of Black Marble) * (Value of Black Marble)

Expected Value = (2/31) * $4 + (6/31) * $2 + (23/31) * (-$1)

Calculating this expression gives us the expected value:

Expected Value = (2/31) * $4 + (6/31) * $2 + (23/31) * (-$1)
= $8/31 + $12/31 - $23/31
= -$3/31

Therefore, if you play this game, your expected value is -$3/31. This means that on average, you are expected to lose approximately $0.10 per game.

2/31 * 4 + 6/31 * 2 - 23/31 = -3/31