# Powerball: Poisson Approximation to Binomial There is a 1/292,201,338 probability of winning the Powerball lottery jackpot with a single ticket. Assume that you purchase a ticket in each of the next 5200 different Powerball games that are run over the next 50 years. Find the probability of winning the jackpot with at least one of those tickets. Is there a good chance that you would win the jackpot at least once in 50 years?

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Powerball: Poisson Approximation to Binomial There is a 1/292,201,338 probability of winning the Powerball lottery jackpot with a single ticket. Assume that you purchase a ticket in each of the next 5200 different Powerball games that are run over the next 50 years. Find the probability of winning the jackpot with at least one of those tickets. Is there a good chance that you would win the jackpot at least once in 50 years?

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Step 1

The probability of winning the Powerball lottery jackpot with a single ticket is 1/292,201,338. Also, a person purchases a ticket in each of the next 5,200 different power ball games that run over the next 50 years.

Requirements for using the Poisson distribution as an approximation to binomial distribution are as follows:

• The number of n trials is greater than or equal to 100, that is, n≥100.
• The mean, np is less than or equal to 10, that is, np≤10.

Where, n is the number of trials and p is the probability of success. If the requirements are satisfied, the Poisson approximation to binomial distribution is used.

Here, the number of trials is 5,200 and is greater than 100. Hence, the first requirement is satisfied.

Step 2

The mean value (µ) is less than 10. Hence, the second requirement is satisfied.

Thus, the Poisson distribution is an approximation to binomial distribution.

Probability of winning with at least one ticket in 50 years:

Define the random variable x as the number of times ...

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