Math challenge - Mega Millions white ball rangePrev TopicNext Topic

• Orange71

  

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  Apr 29, 2023, 11:17 am

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  There are 70 "white" (main) balls in Mega Millions. The player chooses 5 balls from 1 to 70 (or opts for a Quick Pick to randomly choose the same). Define R (for range) as max ball - min ball + 1. For example, if the player chooses the balls {1, 2, 3, 4, 5}, then R = 5. Now consider the case where the balls are chosen randomly, such as a Quick Pick or lottery drawing for the winning numbers. R is now a random variable, and the set of all possible values of R is {1, 2, 3, ... , 70}.

  Suppose the process of randomly choosing 5 balls is repeated infinitely. What are the statistical mean and standard deviation values for R?
• Wavepack

  

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  May 4, 2023, 10:27 pm

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  The possible values are {5, 6, ..., 70}.    Not sure how to solve analytically.   Could simulate to get the answer.
• Orange71

  

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  May 5, 2023, 2:30 pm

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  It can be solved analytically to exact values. No need for a Monte Carlo simulation. If you are interested in the solution I can post it.
• Orange71

  

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  May 6, 2023, 12:30 pm

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  Solution:

  Consider a range, r, as a random variable representing a specific member of the set R = {5,6,7...,70}. The probability of each r is not the same and must be calculated in turn. However, we recognize that all possible random combinations for r = 5,...,70 must add up to ₇₀C₅ = 12103104. This is the fixed denominator in calculating the probability of each r in succession.

  Now let's consider the numerators, which are the numbers of all possible random combinations for each r.  In general, for any given r, for a fixed minimum ball position in the range, e.g. 1, 2, 3, etc., there will be a fixed maximum position. So the number of possible random combinations for these fixed min and max values is _r-2C₃  since there are 5 balls picked. Furthermore, for each r  there are 70 - r + 1 positions for the minimum ball. Therefore, the total number of possible random combinations for a given value of r is _r-2C₃ × (70 - r + 1). The probability of r, which we define as P(r) , is consequently _r-2C₃ × (70 - r + 1)  ÷ ₇₀C_{5 .}

  We calculate the mean of r as μ  = ∑[r × P(r)]. μ in this case is 145/3 or 48 1/3. The standard deviation is σ = {∑[(r - μ)²× P(r)]}^1/2. The standard deviation in this case is 12.104.
• Wavepack

  

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  May 9, 2023, 3:24 am

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  Thanks for posting that counting proof. 

  Looking at a plot of the pdf, the mode looks to be 54.

   

  
• Orange71

  

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  May 11, 2023, 7:54 pm

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  Indeed, that is correct. 54 is the Mode with 375700 combinations out of 12103014 total possible combinations, or around 3.1%.