Suppose that you are offered the following "deal." You roll a six-sided die. If you roll a 6, you win $10. If you roll a 4 or 5, you win $1. Otherwise, you pay $5.

a. Complete the PDF Table. List the X values, where X is the profit, from smallest to largest. Round to 4 decimal places where appropriate.

Probability Distribution Table
X P(X)
-5 ?
1 ?
10 ?
b. Find the expected profit.

Prob(a 6) = 1/6

expected return on this roll = (1/6)($10) = $ 5/3
prob(4or5) = 1/3
expected return on this roll = $1(1/3) = $ 1/3
prob(remaining outcomes) = 3/6 = 1/2
expected return of this outcome = -5(1/2) = -$ 5/2

expected profit or loss = $( 5/3 + 1/3 - 5/2)
= - $ 1/2

Place my results in the appropriate parts of your table

I wouldn't play this game.

a. Here is the completed PDF Table:

Probability Distribution Table
X P(X)
-5 ?
1 ?
10 ?

To find the probabilities for each outcome, we need to determine the probability associated with each X value:

For X = -5: To obtain this outcome, you need to roll a 1, 2, 3, or a 6.
P(X = -5) = P(1) + P(2) + P(3) + P(6) = 4/6 = 2/3

For X = 1: To obtain this outcome, you need to roll a 4 or a 5.
P(X = 1) = P(4) + P(5) = 2/6 = 1/3

For X = 10: To obtain this outcome, you need to roll a 6.
P(X = 10) = P(6) = 1/6

Probability Distribution Table
X P(X)
-5 2/3
1 1/3
10 1/6

b. In order to find the expected profit, we need to multiply each X value by its corresponding probability and sum them up:

Expected profit = (-5)(2/3) + (1)(1/3) + (10)(1/6)
Expected profit = -10/3 + 1/3 + 10/6
Expected profit = (-10 + 1 + 10)/6
Expected profit = (1/6)

So, the expected profit is approximately 0.1667.

a. To complete the PDF table, we need to calculate the probability of each outcome and list them in ascending order of profit (X). Let's calculate the probabilities:

X = -5 (Losing $5)
The probability of rolling any number other than 4, 5, or 6 is 3/6 = 1/2.
P(X = -5) = 1/2

X = 1 (Winning $1)
The probability of rolling either a 4 or a 5 is 2/6 = 1/3.
P(X = 1) = 1/3

X = 10 (Winning $10)
The probability of rolling a 6 is 1/6.
P(X = 10) = 1/6

Now, we can complete the PDF table:

Probability Distribution Table
X P(X)
-5 1/2
1 1/3
10 1/6

b. To find the expected profit, we multiply each profit value by its corresponding probability and then add all the values together. Let's calculate it:

E(X) = (-5)*(1/2) + (1)*(1/3) + (10)*(1/6)
E(X) = -5/2 + 1/3 + 10/6
E(X) = (-15/6) + (2/6) + (10/6)
E(X) = -3/6
E(X) = -1/2

The expected profit for this deal is -$0.50 or -50 cents.

To complete the probability distribution table, we need to determine the probability of each value of X occurring. The probabilities can be calculated using the following formula:

P(X) = (Number of ways X can occur) / (Total number of possible outcomes)

a. Let's calculate the probability for each value of X.

For X = -5:
Rolling any number other than 4, 5, or 6 results in a loss of $5. As there are three outcomes (1, 2, 3) that lead to a loss of $5, and the total number of outcomes is 6, the probability of X = -5 is:

P(X = -5) = 3 / 6 = 0.5

For X = 1:
Rolling a 4 or 5 results in a profit of $1. There are two outcomes (4, 5) that result in a profit of $1, so the probability of X = 1 is:

P(X = 1) = 2 / 6 = 0.3333 (rounded to 4 decimal places)

For X = 10:
Rolling a 6 results in a profit of $10. As there is only one outcome (6) that leads to a profit of $10, the probability of X = 10 is:

P(X = 10) = 1 / 6 = 0.1667 (rounded to 4 decimal places)

Now let's fill in the PDF table:

X | P(X)
-5 | 0.5000
1 | 0.3333
10 | 0.1667

b. To find the expected profit, we multiply each value of X by its corresponding probability, and then sum up the products. Using the formula:

Expected Profit = Σ(X * P(X))

Expected Profit = (-5 * 0.5000) + (1 * 0.3333) + (10 * 0.1667)

Expected Profit = -2.5000 + 0.3333 + 1.6667

Expected Profit = -0.5

Therefore, the expected profit from this deal is -$0.50.