Powerball! The Powerball lottery picks five different integers between 1 and 59
ID: 3817003 • Letter: P
Question
Powerball! The Powerball lottery picks five different integers between 1 and 59 inclusive, and a sixth integer between 1 and 39, which may duplicate one of the earlier five integers. A participant of the lottery wins the jackpot if the first five numbers she picked match the first five numbers drawn (regardless of the order), and the sixth number she picked matches the sixth number drawn. a. How many different tickets can a person buy? For the remaining subproblems, assume that the person just bought one ticket. b. What is the probability her sixth number matches the sixth number drawn? c. What is the probability that her first five numbers matches the first five numbers drawn, but her sixth number doesn't match the sixth number drawn? d. What is the probability that the player matches exactly three of the first five numbers drawn and the sixth number drawn? e. Finally, what is the probability that she wins the jackpot?Explanation / Answer
Answer:
a. First 5 integers between 1 and 59 can be selected in 59P5 ways. 6th integer between 1 and 39 can be selected in 39P1 ways. Since 6th integer can a duplicate of earlier 5 integers, such a number from earlier 5 numbers can be selected in 5P1 ways.
Hence, total number of possible ways = 59P5 * 39P1 * 5P1 = (59!/(59-5)!) * (39!/(39-1)!) * (5!/(5-1)!)
= (59*58*57*56*55*54!/(54)!) * (39*38!/38!) * (5*4!/4!) = 59*58*57*56*55*39*5 = 117149432400
This number express the number of tickets a person can buy.
b. Sixth number drawn will be any number between 1 and 39. The probability of selecting such a number will be 1/39. As the person has only one ticket, the probability of sixth integer on ticket matching with sixth integer drawn will be 1/39. Here it is to be noted that it is not mentioned in the problem that only sixth integer is to be matched and first 5 integers are not be matched.
c. As said above first 5 integers can be selected in 59P5 = (59!/(59-5)!) = (59*58*57*56*55*54!/(54)!) = 59*58*57*56*55 = 600766320 ways. The probability of selecting such 5 integers will be =
no. of ways of selecting 5 integers / total number of ways = 600766320/117149432400
The probability that sixth integer does not match will be 38/39.
Hence overall probability will be = (600766320/117149432400)*(38/39) = 0.00499671268
d. First 3 intergers can be selected in 59*58*57 = 195054
Probability of selecting such three integers will be = 195054/117149432400
Probability of selecting 6th integer is 1/39.
Hence overall probability = (195054/117149432400)*(1/39) = 4.269235038465808e-8
e. She is having one ticket only. She will win jackpot iff first 5 integers match with first integers drawn and sixth integer matches with sixth integer drawn. This is possible only in 1 way.
Hence probability of winning jackpot = 1/117149432400 = 8.5361062e-12
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