The Mega Millions game consists of drawing five numbers from the integers 1,2,3,

Question

The Mega Millions game consists of drawing five numbers from the integers 1,2,3,...,70 (without replacement). Then a special number, a sixth number, is selected froma new set of numbers 1,2,3,...,25. A winning player must have selected the correct five numbers from the first set and the correct number from the second set. You get one set of six numbers for a $2 bet. Please assume that 1.2 billion is wagered (i.e, 600 million tickets are pruchased).

What is the probability that there will be no winners?

What is the probability there will be exactly one winner?

What is the probability that there will be two or more winners?

What is the expected number of winners?

What is the median nmber of winners?

What is the modal number of winners?

Explanation / Answer

Here first we will find the probability of win here for any random ticket owner = 1/ 70C5 * 25 = 1/302575350

so, here total tickets = 600 million = 600000000

Pt(No winners) = 600000000C0 * (1/302575350)0 * (302575349/302575350)600000000 = 0.1377

Here we can actually solve it with the help of poisson distribution as n is too high and p is too low.

so Expected number of winners = 600000000 * (1/302575350) = 1.983

Here,

Pr(0 winner) = e-1.983 1.9830/0! = 0.1377

Pr(exactly 1 winner) = e-1.983 1.9831/1! = 0.2730

Pr(2 or More than 2 winner) = 1 - Pr(0 winner) - Pr(1 winner) = 1 - 0.1377 - 0.2730 = 1 - 0.4107 = 0.5893

Here median number of winners will be the value for which cumulative probability would be 0.5.

so here we can see that Pr(0 or 1 winner) = 0.4107 so it is obvious that median number of winners would be 2. Similarly, mean value is very near to the value of 2 so modal number of winners shall also be 2.

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