Question

Imagine that the State of Kentucky want s to create a new lottery similar to the PowerBall and Mega Millions that has better odds of winning. Kentucky assumes better odds of winning would draw more entries from both Kentucky and surrounding states because of the better odds of winning millions of dollars. Assume Kentucky plans to use 50 white balls and 30 red balls. In this mythical lottery, what is the probability with one entry of matching the five white balls drawn in any order (when a ball is drawn it is not replaced) and matching the one red ball drawn for the grand prize? Assume five white balls are randomly drawn out of a drum, there are fifty balls numbered from one to fifty. Then assume that one red ball is randomly drawn out of a drum, there are 30 red balls numbered from one to thirty.

Answer 1

Since these are drawn without replacement, these numbers cannot be repeated. Suppose the 5 winning numbers are 11 55 42 30 and 15, these can come in any order in 5! ways = 120 ways (The rule of permutation of n distinct thing all taken together = n!)

Probability For the 5 white Numbers: (lets take 1 order as given above 11 55 42 30 and 15, then multiply the probability by 120 for drawing 5 correct white numbers)

P(drawing a 11 out of 50 available balls) = 1/50

P(drawing a 55 out of 49 available balls) = 1/49

P(drawing a 42 out of 50 available balls) = 1/48

P(drawing a 30 out of 50 available balls) = 1/47

P(drawing a 15 out of 50 available balls) = 1/46

Therefore the probability of drawing the 5 correct numbers = 120 * (1/50) * (1/49) * (1/48) * (1/47) * (1/46) = 1/12118760

P(of picking the correct red ball, from a drum containg 30 numbers) = 1/30

Therefore the required probability = (1/2118760) * (1/30) = 1/63562800 = 1.57325 * 10^-8.