A fair six-sided die is rolled. If the number showing is even, you lose a point for each dot showing. If the number showing is odd, you win a point for each dot showing.
a. Find E(x)for one roll if x represents the number of point s you win.
b. Find the expected winnings for ten rolls.
To find the expected value for one roll, we need to multiply the probability of each outcome by its corresponding value (number of points won or lost) and sum them up.
a. Expected value for one roll, E(x):
Let's denote the value on the die as X. Since it is a fair six-sided die, each outcome has a probability of 1/6.
Possible outcomes for X: {1, 2, 3, 4, 5, 6}
For even numbers (X=2, 4, 6): You lose a point equal to the number showing (-2, -4, -6).
For odd numbers (X=1, 3, 5): You win a point equal to the number showing (1, 3, 5).
Calculating E(x):
E(x) = (1/6)*(-2) + (1/6)*(-4) + (1/6)*(-6) + (1/6)*(1) + (1/6)*(3) + (1/6)*(5)
E(x) = -1/3 - 2/3 - 1 + 1/6 + 1/2 + 5/6
E(x) = -2/3 + 2/6 + 1/2 + 5/6
E(x) = -2/3 + 1/3 + 5/6
E(x) = (-4 + 2 + 5)/6
E(x) = 3/6
E(x) = 1/2
Therefore, the expected number of points won for one roll is 1/2.
b. To find the expected winnings for ten rolls, we multiply the expected value for one roll by the number of rolls (10):
Expected winnings for ten rolls = 10 * E(x)
Expected winnings for ten rolls = 10 * 1/2
Expected winnings for ten rolls = 10/2
Expected winnings for ten rolls = 5
Therefore, the expected winnings for ten rolls is 5 points.
To find the expected value, denoted as E(x), we need to multiply each possible outcome by its corresponding probability, and then sum up these products.
In this case, let's consider a single roll of the die. The possible outcomes are the numbers 1, 2, 3, 4, 5, and 6.
a. Find E(x) for one roll:
For each outcome, we determine the number of points won or lost. If the outcome is even, we lose a point for each dot (the number showing). If the outcome is odd, we win a point for each dot.
Number 1: Odd, win 1 point
Number 2: Even, lose 2 points
Number 3: Odd, win 3 points
Number 4: Even, lose 4 points
Number 5: Odd, win 5 points
Number 6: Even, lose 6 points
Now we need to calculate the probabilities of each outcome:
The probability of getting each number on a fair six-sided die is 1/6, since there are 6 equally likely outcomes.
Therefore, the expected value for this single roll (E(x)) is:
E(x) = (1/6) * 1 + (1/6) * (-2) + (1/6) * 3 + (1/6) * (-4) + (1/6) * 5 + (1/6) * (-6)
Simplifying, we get:
E(x) = (1 - 2/6 + 3/6 - 4/6 + 5/6 - 6/6)
E(x) = (-3/6) = -0.5
b. Find the expected winnings for ten rolls:
To calculate the expected winnings for ten rolls, we can multiply the expected value of a single roll (E(x)) by 10. This is because the expected value is linear, meaning that the expected value of a sum of random variables is equal to the sum of their individual expected values.
Expected winnings for ten rolls = 10 * E(x) = 10 * (-0.5) = -5
Therefore, the expected winnings for ten rolls would be -5. This means that, on average, over many sets of ten rolls, you can expect to lose 5 points in total.