# Probability and the Win 4 Lottery

In the New York State Win 4 lottery, you place a bet by selecting four digits. Repetition is allowed, and winning requires that your sequence of four digits matches the four digits that are later drawn. Assume you placed one bet with a sequence of four digits.

a. Use the multiplication rule to find the probability that your first two digits match those drawn and your last two digits do not match those drawn. That is, find P (MMXX), where M denotes a match and X denotes a digit that does not match the winning number.

b. Beginning with MMXX, make a complete list of the different possible arrangements of two matching digits and two digits that do not match, then find the probability for each entry in the list.

c. Based on the preceding results, what is the probability of getting exactly two matching digits when you select four digits for the Win 4 lottery game?

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#### Solution Summary

The solution comprises of detailed step-by-step calculations using the multiplication rule to find the probability of winning the lottery.

Gender of Children, Birth Dates, Kentucky Lottery

See attached file.

You may use a calculator on this assignment. Please show work where applicable as this will help me when assigning partial credit.

Probability Questions

1. Find the probability of a couple having at least 1 girl among 7 children. Assume that boys and girls are equally likely and that the gender of a child is independent of any other child.

2. If the couple has seven children and they are all boys, what can the couple conclude?

3. Find the probability that a randomly selected subject has a birthday on the 26th of the month, given that the subject is born in June. That is, find P(Birthday on the 26th | June birthday).

Assume we are dealing with a 365 day year.

Birthday on the 26th Birthday not on the 26th

Birthday in June 1 29

Birthday not in June 11 324

4. Find the probability that a randomly selected subject has a birthday in June given that the subject was born on the 26th. That is find P(June birthday | Birthday on the 26th). Assume we are dealing with a 365 day year.

5. The Kentucky Lottery has a Pick 4 lottery game, you pay $1 to select a sequence of four digits, such as 1332. If you select the same sequence of four digits that are drawn, you have a "straight" match and you win $5000.

a) How many different selections are possible?

b) What is the probability of winning?

c) If you win, what is your net profit?

d) Find the expected value.

x, new profit P(x), Probability x . P(x)

Win

Lose

E=σ[ x⋅P( x)]=