A bag contains 1 gold marbles, 10 silver marbles, and 25 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?
Well, let's calculate the expected value together, shall we?
The probability of selecting a gold marble is 1/36 (1 gold out of 36 total marbles).
So, if you win $3 with a probability of 1/36, your expected value from gold marbles is: (1/36) * $3 = $0.08.
The probability of selecting a silver marble is 10/36 (10 silver out of 36 total marbles).
So, if you win $2 with a probability of 10/36, your expected value from silver marbles is: (10/36) * $2 = $0.56.
The probability of selecting a black marble is 25/36 (25 black out of 36 total marbles).
So, if you lose $1 with a probability of 25/36, your expected value from black marbles is: (25/36) * (-$1) = -$0.69.
Now, let's add up the expected values from each type of marble:
$0.08 + $0.56 - $0.69 = -$0.05.
So, based on the probabilities and payouts, the expected value of playing this game is -$0.05.
But remember, when it comes to gambling, it's always good to expect the unexpected!
To find the expected value, we need to multiply each possible outcome by its corresponding probability and sum them up.
Let's calculate the expected value step-by-step:
Step 1: Calculate the probability of selecting each color marble:
- The probability of selecting a gold marble = Number of gold marbles / Total number of marbles = 1 / (1 + 10 + 25) = 1 / 36
- The probability of selecting a silver marble = Number of silver marbles / Total number of marbles = 10 / 36
- The probability of selecting a black marble = Number of black marbles / Total number of marbles = 25 / 36
Step 2: Calculate the expected value of each color marble:
- For gold marble: Probability of gold marble * Value of winning = (1 / 36) * $3 = $1/12
- For silver marble: Probability of silver marble * Value of winning = (10 / 36) * $2 = $5/9
- For black marble: Probability of black marble * Value of losing = (25 / 36) * (-$1) = -$25/36
Step 3: Sum up the expected values:
Expected value = Expected value of gold marble + Expected value of silver marble + Expected value of black marble
= $1/12 + $5/9 + (-$25/36) = ($3 + $20 - $25) / 36 = -$2/36 = -$1/18
Therefore, the expected value of playing this game is -$1/18, which means you can expect to lose an average of $1/18 every time you play.
To calculate the expected value, we need to multiply the probability of each outcome by the corresponding value and then sum them up.
First, let's determine the probabilities of selecting each type of marble from the bag:
P(gold) = number of gold marbles / total number of marbles = 1 / (1 + 10 + 25) = 1 / 36
P(silver) = number of silver marbles / total number of marbles = 10 / 36
P(black) = number of black marbles / total number of marbles = 25 / 36
Now, let's calculate the expected value:
E(gold) = P(gold) * value of gold = (1 / 36) * $3 = $0.0833...
E(silver) = P(silver) * value of silver = (10 / 36) * $2 = $0.5555...
E(black) = P(black) * value of black = (25 / 36) * (-$1) = -$0.6944...
Expected value = E(gold) + E(silver) + E(black)
= $0.0833... + $0.5555... + (-$0.6944...)
= -$0.0556...
Therefore, the expected value of playing this game is approximately -$0.0556.
This means that, on average, you can expect to lose about $0.056 per game if you repeatedly play this game over a large number of trials.
multiply each outcome by its probability, and add them up. That gives you
(1/36 * 3) + (10/36 * 2) + (25/36 * -1) = -1/18
As expected, the house has the edge.