Math Problem Statement
Billy purchases one 6-49 lottery ticket every week and keeps track of the number of “matches”” he has on each of his tickets. To be clear, a “match” will occur when a number on his ticket matches a number that appears in the winning combination. A random variable 𝑋 that keeps track of the number of matching numbers Billy experiences per week has the probability distribution function with a mean and standard deviation of 𝑃(𝑋=𝑥)=(6𝑥)(436−𝑥)(496)𝑥=0,1,2,3,4,5,6.𝐸(𝑋)=𝜇𝑋=3649=0.7347𝑆𝐷(𝑋)=𝜎𝑋=0.75998≈0.76 Billy claims that in a year (52 weeks), on average, he manages to have at least one matching number on his 6-49 ticket. What do you think about Billy’s claim? Provide a brief commentary about Billy’s claim using your current knowledge of statistics and probability theory.
Solution
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Math Problem Analysis
Mathematical Concepts
Probability
Random Variables
Expected Value
Standard Deviation
Binomial Distribution
Formulas
P(X=x) = (6 choose x)(43 choose 6-x) / (49 choose 6)
E(X) = μ_X = 0.7347
SD(X) = σ_X = 0.75998
Theorems
Law of Large Numbers
Binomial Distribution
Suitable Grade Level
Undergraduate level (Probability and Statistics)
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