How Do You Calculate Expected Value in a Dice Game?

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SUMMARY

The expected value calculation for the dice game involving two dice reveals that players can expect to lose approximately $1.67 per game. The game structure awards $5 for rolling a 2, 3, 4, 10, 11, or 12, while it penalizes players with a $5 loss for rolling a 5, 6, 7, 8, or 9. The completed probability table shows the net gain/loss and the product of net gain/loss and probability for each possible outcome, leading to the final expected value of -$5/3.

PREREQUISITES
  • Understanding of basic probability concepts
  • Familiarity with expected value calculations
  • Ability to construct and interpret probability tables
  • Knowledge of dice game mechanics
NEXT STEPS
  • Study advanced probability theory, focusing on expected value in games of chance
  • Learn about variance and standard deviation in probability distributions
  • Explore simulations of dice games using Python or R for practical applications
  • Investigate other gambling games and their expected values for comparative analysis
USEFUL FOR

Mathematicians, game designers, statisticians, and anyone interested in probability theory and expected value calculations in gaming scenarios.

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:

$$\text{EP}=-\frac{5}{3}$$
 

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