Anticipated Value in Roulette

Solution & Explanation

Determining the anticipated or expected value of a roulette game with numbers ranging from 1 to 32 involves understanding the concept of the Discrete Uniform Distribution. In this distribution, each outcome is equally likely, and the expected value can be calculated using a straightforward formula.

Step-by-step Calculation:

1. Understanding the Roulette Setup:

   • The roulette wheel has numbers ranging from 1 to 32.
   • Each number is equally likely to be selected when the wheel is spun.

2. Defining the Discrete Uniform Distribution:

   • In a discrete uniform distribution, each of the n outcomes has an equal probability of occurring. For our roulette wheel:
     • The minimum value (a) is 1.
     • The maximum value (b) is 32.
     • The number of possible outcomes (n) is 32.

3. Calculating the Expected Value (E[X]):

   • The formula for the expected value of a discrete uniform distribution is given by:

     $E[X] = \frac{a + b}{2}$

   • Substituting the values from our roulette wheel:

     $E[X] = \frac{1 + 32}{2}$

   • Simplifying the calculation:

     $E[X] = \frac{33}{2} = 16.5$

4. Interpreting the Result:

   • The expected value of 16.5 represents the average number you would expect to land on if the roulette wheel were spun an infinite number of times.
   • It is the midpoint of the distribution, as each number from 1 to 32 is equally likely to occur.

5. Alternative Verification:

   • Another way to verify this is by calculating the sum of all possible outcomes and dividing by the total number of outcomes:

     $E = \frac{1}{32} \times (1 + 2 + 3 + ... + 32)$

   • The sum of numbers from 1 to 32 can be calculated using the formula for the sum of an arithmetic series:

     $S = \frac{n}{2} \times (a + b) = \frac{32}{2} \times (1 + 32) = 528$

   • Thus, the expected value is:

     $E = \frac{528}{32} = 16.5$

Conclusion

The expected value of a roulette game with numbers ranging from 1 to 32 is 16.5. This value is derived from the principles of the discrete uniform distribution, where each number has an equal chance of being selected, and it represents the average outcome over many spins of the wheel.