Question

section 6.1 homework score: 8.67/21 answered: 9/21 question 12 suppose that you are offered the following \deal.\ you roll a six - sided die. if you roll a 2, 3, 4 or 5, you win $5. otherwise, you pay $10. if you roll a 6, you win $7. 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) b. find the expected profit. $ - 5.50 (round to the nearest cent) c. interpret the expected value. if you play many games you will likely win on average very close to $2.83 per game. this is the most likely amount of money you will win. you will win this much if you play a game.

Explanation:

Step1: Determine the probabilities and profits

The probability of rolling each number on a fair six - sided die is $P=\frac{1}{6}$.
When $X = 7$ (rolling a 6), $P(X = 7)=\frac{1}{6}$; when $X = 5$ (rolling 2, 3, 4, or 5), the number of favorable outcomes is 4, so $P(X = 5)=\frac{4}{6}$; when $X=- 10$ (rolling a 1), $P(X=-10)=\frac{1}{6}$.

Step2: Calculate the expected value formula

The formula for the expected value $E(X)=\sum_{i}x_{i}P(x_{i})$.
$E(X)=7\times\frac{1}{6}+5\times\frac{4}{6}+(-10)\times\frac{1}{6}$.

Step3: Perform the calculations

$E(X)=\frac{7 + 20-10}{6}=\frac{17}{6}\approx2.83$.
The cost of playing is $10$. So the expected profit is $E = 2.83-10=-7.17$. But if we consider the non - cost adjusted expected value of the pay - out, we have:
$E(X)=7\times\frac{1}{6}+5\times\frac{4}{6}+(-10)\times\frac{1}{6}=\frac{7 + 20 - 10}{6}=\frac{17}{6}\approx2.83$. After subtracting the cost of $10$ to play, the expected profit is $2.83-10=-7.17$.

a. The probability distribution table:

$X$	$P(X)$
5	$\frac{4}{6}\approx0.6667$
7	$\frac{1}{6}\approx0.1667$

b. The expected profit:
The expected value of the pay - out is $\sum_{x}xP(x)=7\times\frac{1}{6}+5\times\frac{4}{6}+(-10)\times\frac{1}{6}=\frac{7 + 20-10}{6}=\frac{17}{6}\approx2.83$. Since the cost to play is $10$, the expected profit is $2.83 - 10=-7.17\approx - 7.17$. Rounding to the nearest cent, the expected profit is $-\$7.17$.

c. Since the expected profit is negative ($-\$7.17$), you should not play this game.

Answer:

a.

$X$	$P(X)$
5	$0.6667$
7	$0.1667$

b. $-\$7.17$
c. No.