MHB How Do You Calculate Expected Value in a Dice Game?

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In the discussed dice game, rolling a 2, 3, 4, 10, 11, or 12 results in a win of $5, while rolling a 5, 6, 7, 8, or 9 results in a loss of $5. A probability table was constructed to calculate the expected value, showing the net gain or loss associated with each possible outcome. After completing the table and summing the products of gain/loss and their probabilities, the expected value was determined to be -$5/3. This indicates that, on average, a player can expect to lose approximately $1.67 per game.
AnnBean
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I really need help on how to solve this (Sweating):

A dice game involves rolling 2 dice. If you roll a 2, 3 , 4, 10, 11 or a 12, you win \$5. If you roll a 5, 6, 7, 8, or 9, you lose \$5. Find the expected value you win (or lose) per game.
 
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Re: Probability

Hello, and welcome to MHB, AnnBean! (Wave)

I would begin by constructing a table as follows:

[table="width: 800, class: grid, align: left"]
[tr]
[td]Sum $S$[/td]
[td]Probability of $S$: $P(S)$[/td]
[td]Net Gain/Loss (in dollars) $G$[/td]
[td]Product $G\cdot P(S)$[/td]
[/tr]
[tr]
[td]2[/td]
[td]$\dfrac{1}{36}$[/td]
[td]5[/td]
[td]$\dfrac{5}{36}$[/td]
[/tr]
[tr]
[td]3[/td]
[td]$\dfrac{1}{18}$[/td]
[td]5[/td]
[td]$\dfrac{5}{18}$[/td]
[/tr]
[/table]

Can you complete the table?
 
Re: Probability

Just to follow up, here is the completed table:

[table="width: 800, class: grid, align: left"]
[tr]
[td]Sum $S$[/td]
[td]Probability of $S$: $P(S)$[/td]
[td]Net Gain/Loss (in dollars) $G$[/td]
[td]Product $G\cdot P(S)$[/td]
[/tr]
[tr]
[td]2[/td]
[td]$\dfrac{1}{36}$[/td]
[td]5[/td]
[td]$\dfrac{5}{36}$[/td]
[/tr]
[tr]
[td]3[/td]
[td]$\dfrac{1}{18}$[/td]
[td]5[/td]
[td]$\dfrac{5}{18}$[/td]
[/tr]
[tr]
[td]4[/td]
[td]$\dfrac{1}{12}$[/td]
[td]5[/td]
[td]$\dfrac{5}{12}$[/td]
[/tr]
[tr]
[td]5[/td]
[td]$\dfrac{1}{9}$[/td]
[td]-5[/td]
[td]$-\dfrac{5}{9}$[/td]
[/tr]
[tr]
[td]6[/td]
[td]$\dfrac{5}{36}$[/td]
[td]-5[/td]
[td]$-\dfrac{25}{36}$[/td]
[/tr]
[tr]
[td]7[/td]
[td]$\dfrac{1}{6}$[/td]
[td]-5[/td]
[td]$-\dfrac{5}{6}$[/td]
[/tr]
[tr]
[td]8[/td]
[td]$\dfrac{5}{36}$[/td]
[td]-5[/td]
[td]$-\dfrac{25}{36}$[/td]
[/tr]
[tr]
[td]9[/td]
[td]$\dfrac{1}{9}$[/td]
[td]-5[/td]
[td]$-\dfrac{5}{9}$[/td]
[/tr]
[tr]
[td]10[/td]
[td]$\dfrac{1}{12}$[/td]
[td]5[/td]
[td]$\dfrac{5}{12}$[/td]
[/tr]
[tr]
[td]11[/td]
[td]$\dfrac{1}{18}$[/td]
[td]5[/td]
[td]$\dfrac{5}{18}$[/td]
[/tr]
[tr]
[td]12[/td]
[td]$\dfrac{1}{36}$[/td]
[td]5[/td]
[td]$\dfrac{5}{36}$[/td]
[/tr]

[/table]

Summing up the $G\cdot P(S)$, column, we find the expected value is:

$$\text{EP}=-\frac{5}{3}$$
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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