If a player rolls 2 dice and gets a sum of 3 or 11 he collects $15. if he gets a sum of 12 he gets $3. the cost to play the game is $2. let x be the net money gained by the player.
a. construct the prob distribution of the random variable x
b. find the expected net/gain loss of the player
How would i do this?
To solve this problem, we need to analyze the possible outcomes when rolling two dice. Each outcome consists of a pair of numbers, and the sum of those numbers determines the net gain or loss for the player.
a. Constructing the probability distribution:
We'll calculate the probability for each possible sum of the two dice.
There are 36 possible outcomes when rolling two dice, as each die has 6 possible numbers.
For a sum of 3:
There is only one outcome: (1, 2). The player gains $15.
P(sum of 3) = 1/36
For a sum of 11:
There are two possible outcomes: (5, 6) and (6, 5). The player gains $15 for each of these outcomes.
P(sum of 11) = 2/36 = 1/18
For a sum of 12:
There is only one outcome: (6, 6). The player gains $3.
P(sum of 12) = 1/36
For any other sum, the player gains nothing.
P(any other sum) = 1 - (P(sum of 3) + P(sum of 11) + P(sum of 12)) = 1 - (1/36 + 1/18 + 1/36) = 1 - 1/12 = 11/12
Now we can construct the probability distribution:
(Net Gain/Loss) (Probability)
$15 1/36
$15 1/18
$3 1/36
$0 11/12
b. Finding the expected net/gain loss:
To find the expected net/gain loss, we multiply each outcome by its corresponding probability and then sum these values.
Expected net/gain loss = ($15 * 1/36) + ($15 * 1/18) + ($3 * 1/36) + ($0 * 11/12)
Calculate each term separately:
($15 * 1/36) = $15/36 = $5/12
($15 * 1/18) = $15/18 = $5/6
($3 * 1/36) = $3/36 = $1/12
($0 * 11/12) = $0
Now sum these values:
Expected net/gain loss = $5/12 + $5/6 + $1/12 + $0 = $1/2 + $5/6 = $3/6 + $5/6 = $8/6 = $4/3
Therefore, the expected net/gain loss of the player is $4/3.
To construct the probability distribution of the random variable x, we need to calculate the probability of each possible outcome and the corresponding net money gained by the player.
a. Probability distribution of the random variable x:
Let's consider the possible outcomes of rolling two dice:
- Sum of 3: There is only one combination possible, which is rolling a 1 on each dice. The probability of this outcome is 1/36.
- Sum of 11: There are two combinations possible, which are rolling a 5 and a 6, or rolling a 6 and a 5. The probability of this outcome is 2/36.
- Sum of 12: There is only one combination possible, which is rolling a 6 on each dice. The probability of this outcome is 1/36.
- Any other sum: This includes sums from 4 to 10. There are multiple combinations possible for each sum, so let's calculate the probability of each:
- Sum of 4: There are three combinations possible, which are rolling a 1 and a 3, a 2 and a 2, or a 3 and a 1. The probability of this outcome is 3/36.
- Sum of 5: There are four combinations possible, which are rolling a 1 and a 4, a 2 and a 3, a 3 and a 2, or a 4 and a 1. The probability of this outcome is 4/36.
- Sum of 6: There are five combinations possible, which are rolling a 1 and a 5, a 2 and a 4, a 3 and a 3, a 4 and a 2, or a 5 and a 1. The probability of this outcome is 5/36.
- Sum of 7: There are six combinations possible, which are rolling a 1 and a 6, a 2 and a 5, a 3 and a 4, a 4 and a 3, a 5 and a 2, or a 6 and a 1. The probability of this outcome is 6/36.
- Sum of 8: There are five combinations possible, which are rolling a 2 and a 6, a 3 and a 5, a 4 and a 4, a 5 and a 3, or a 6 and a 2. The probability of this outcome is 5/36.
- Sum of 9: There are four combinations possible, which are rolling a 3 and a 6, a 4 and a 5, a 5 and a 4, or a 6 and a 3. The probability of this outcome is 4/36.
- Sum of 10: There are three combinations possible, which are rolling a 4 and a 6, a 5 and a 5, or a 6 and a 4. The probability of this outcome is 3/36.
Now we can calculate the net money gained by the player for each outcome:
- For a sum of 3 or 11, the player collects $15, so the net money gained is $15 - $2 = $13.
- For a sum of 12, the player collects $3, so the net money gained is $3 - $2 = $1.
- For any other sum, the player loses the $2 cost to play, so the net money gained is -$2.
Therefore, the probability distribution of the random variable x is as follows:
x | Probability
----------------------
13 | 1/36 + 2/36
1 | 1/36
-2 | 3/36 + 4/36 + 5/36 + 6/36 + 5/36 + 4/36 + 3/36
b. To find the expected net gain/loss of the player, we multiply each net money gained by its corresponding probability and sum them up:
Expected net gain/loss = (13 * (1/36 + 2/36)) + (1 * 1/36) + (-2 * (3/36 + 4/36 + 5/36 + 6/36 + 5/36 + 4/36 + 3/36))
Simplifying the equation gives you the expected net gain/loss of the player.