On September 11, 2002, a particular state lottery's daily number came up 9-1-1. Assume that no more than one digit is used to represent the first nine months. a) What is the probability that the winning three numbers match the date on any given day? b) What is the probability that a whole year passes without this happening? c) What is the probability that the date and winning lottery number match at least once during any year? d) If 27 states have a three-digit lottery, what is the probability that at least one of them will come up 1-1-4 on January 14?

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The

probabilitythat thewinning three numbersmatch the date on any given day is thelikelihoodof thewinning numberprobabilitythat thewinning three numbersmatch the date on anygiven dayis 0.001probabilitythat a whole year passes without this happening is 0.7407probabilitythat the date andwinning lottery numbermatchat least onceduring any year is 0.2593probabilitythatat least oneof them will come up 1-1-4 on January 14 is 0.0267## The probability that the winning three numbers match the date on any given day

Start by calculating the number of

possible outcomesEach of the

three digitscan take any of 0 - 9 i.e. 10 digits.So, the number of

possible outcomesis:There can be only one winning date.

So, the

probability(p) is:This gives

Hence, the

probabilitythat thewinning three numbersmatch the date on anygiven dayis 0.001## The probability that a whole year passes without this happening

In a calendar year, there are 65 days, where the whole year passes without

matchingthewinning numbers.The dates are:

This means that the

matching numberscan occur on the remaining 300 days.The

probabilitythat thewinning datedoes not occur on a date is calculated using the followingcomplement ruleSo, we have:

For the 300 days, we have:

Hence, the

probabilitythat a whole year passes without this happening is 0.7407## The probability that the date and winning lottery number match at least once during any year

This is calculated using the following

complement ruleSo, we have:

Hence, the

probabilitythat the date andwinning lottery numbermatchat least onceduring any year is 0.2593## The probability that at least one of them will come up 1-1-4 on January 14

In (b), we have:

So, the

probabilityis calculated as:Where n is the number of states i.e. n = 27.

So, we have:

Hence, the

probabilitythatat least oneof them will come up 1-1-4 on January 14 is 0.0267Read more about

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