Math Problem Statement

Determine my theoretical probability and expected values for this: probability is Three of a Kind Three of a kind 6/216 10 Consecutive Numbers 12/216 5 Sum Equals 10 27/216 2 Not Winning not sure 0

Solution

Let’s calculate the theoretical probabilities and expected values for the given scenarios. Here's a step-by-step breakdown of each:


1. Understanding the Components

  • Theoretical Probability is the likelihood of each event happening, calculated as P(E)=Favorable OutcomesTotal OutcomesP(E) = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}}.
  • Expected Value (EV) is calculated by multiplying the probability of each event by its corresponding payout: EV=P(E)×PayoutEV = P(E) \times \text{Payout}

2. Calculations

(a) Three of a Kind

  • Probability:
    P(Three of a Kind)=6216=0.0278(or 2.78%)P(\text{Three of a Kind}) = \frac{6}{216} = 0.0278 \, (\text{or } 2.78\%)
  • Expected Value: EV=0.0278×10=0.278EV = 0.0278 \times 10 = 0.278

(b) Consecutive Numbers

  • Probability:
    P(Consecutive Numbers)=12216=0.0556(or 5.56%)P(\text{Consecutive Numbers}) = \frac{12}{216} = 0.0556 \, (\text{or } 5.56\%)
  • Expected Value: EV=0.0556×5=0.278EV = 0.0556 \times 5 = 0.278

(c) Sum Equals 10

  • Probability:
    P(Sum Equals 10)=27216=0.125(or 12.5%)P(\text{Sum Equals 10}) = \frac{27}{216} = 0.125 \, (\text{or } 12.5\%)
  • Expected Value: EV=0.125×2=0.25EV = 0.125 \times 2 = 0.25

(d) Not Winning

  • Probability:
    Since the total probability must sum to 1, calculate P(Not Winning)P(\text{Not Winning}) as: P(Not Winning)=1(6216+12216+27216)=10.2083=0.7917P(\text{Not Winning}) = 1 - \left( \frac{6}{216} + \frac{12}{216} + \frac{27}{216} \right) = 1 - 0.2083 = 0.7917
  • Expected Value: EV=0.7917×0=0EV = 0.7917 \times 0 = 0

3. Summary of Results

OutcomeProbability (PP)Expected Value (EVEV)
Three of a Kind0.0278 (2.78%)0.278
Consecutive Numbers0.0556 (5.56%)0.278
Sum Equals 100.125 (12.5%)0.25
Not Winning0.7917 (79.17%)0

Insights:

  1. The total expected value of the game is: EVtotal=0.278+0.278+0.25+0=0.806EV_{\text{total}} = 0.278 + 0.278 + 0.25 + 0 = 0.806 This means, on average, for every round played, you expect to win 0.806 units.

Would you like additional details on how probabilities or expected values are derived? Here are related questions to explore:


  1. What are the odds of winning any prize in this game?
  2. How does increasing the payout for "Sum Equals 10" change the expected value?
  3. What is the difference between theoretical and experimental probability?
  4. What is the mathematical formula for expected value in general cases?
  5. How can probabilities be visualized effectively for better understanding?

Tip: Always double-check probabilities to ensure they sum up to 1 for accuracy.

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Math Problem Analysis

Mathematical Concepts

Probability
Expected Value

Formulas

P(E) = Favorable Outcomes / Total Outcomes
EV = P(E) × Payout

Theorems

Law of Total Probability

Suitable Grade Level

Grades 9-12