A bag contains 1 gold marbles, 9 silver marbles, and 26 black marbles. Someone offers to play this game: You randomly select one marble from the bag. 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?
Well, well, well! Let's have some fun calculating the expected value of this game.
To do this, we need to multiply the value of each outcome by its corresponding probability and then sum them all up. So, let's crunch some numbers!
The probability of picking a gold marble is 1 out of 36, since there is only one gold marble out of the total 36 marbles in the bag. So, the expected value for a gold marble is (1/36) * $3 = $0.08.
The probability of picking a silver marble is 9 out of 36, giving it a probability of 1/4. Thus, the expected value for a silver marble is (1/4) * $2 = $0.50.
Now, the probability of picking a black marble is 26 out of 36, or 13/18. Thus, the expected value for a black marble is (13/18) * (-$1) = -$0.72.
Now we add up all the expected values: $0.08 + $0.50 - $0.72 = -$0.14.
So, my friend, the expected value of this game is that you would lose $0.14 on average. Oof, tough luck! But remember, it's all just numbers, so don't be too upset.
To calculate the expected value, we need to multiply each outcome by its probability and then sum up the results.
Let's calculate the probability of drawing each type of marble first:
The probability of drawing a gold marble is 1 out of a total of 1 + 9 + 26 = 36 marbles, which can be written as 1/36.
The probability of drawing a silver marble is 9 out of 36 marbles, which can be written as 9/36 or 1/4.
The probability of drawing a black marble is 26 out of 36 marbles, which can be written as 26/36 or 13/18.
Now, let's calculate the expected value:
Expected value = (Probability of winning * Amount won) - (Probability of losing * Amount lost)
Expected value = [(1/36) * $3] + [(9/36) * $2] + [(26/36) * (-$1)]
Expected value = $0.08 + $0.50 - $0.72
Expected value = $0.58 - $0.72
Expected value = -$0.14
Therefore, the expected value if you play this game is -$0.14.
To calculate the expected value, we need to multiply each possible outcome by its respective probability and then sum them up. Let's calculate the expected value step by step.
First, we need to determine the probability of drawing each type of marble. There are a total of 1 + 9 + 26 = 36 marbles in the bag, and only 1 of them is gold. Therefore, the probability of drawing a gold marble is 1/36.
Similarly, there are a total of 9 silver marbles, so the probability of drawing a silver marble is 9/36 or 1/4.
Finally, there are 26 black marbles, so the probability of drawing a black marble is 26/36 or 13/18.
Next, we calculate the expected value by multiplying each outcome by its probability:
E(X) = (3 * 1/36) + (2 * 9/36) + (-1 * 26/36)
E(X) = 3/36 + 18/36 - 26/36
E(X) = -5/36
Therefore, the expected value of playing this game is -5/36, which means that on average, you can expect to lose approximately $0.14 each time you play.
chance of gold = 1/36
chance of silver = 9/36
chance of black = 26/36
1/36 * 3 + 9/36 * 2 - 26/36 * 1
3/36 + 18/36 - 26/36
= -5/36
:(