A state runs a lottery in which six #s are randomly selected from 40 without rep

Question

A state runs a lottery in which six #s are randomly selected from 40 without replacement. A player chooses six #s before the state's sample is selected. (a) What is the probability that the six #s chosen by a player match all six #s in the state's sample? (b) What is the probability that five of the six #s chosen by a player appear in the state's sample? (c) What is the probability that four of the six #s chosen by a player appear in the state's sample? (d) If a player enters one lottery each week. What is the expected # of weeks until a player matches all six #s in the sample?

Explanation / Answer

4a) Total number of combinations is 40C6 = 3838380
Number of ways to match the 6 numbers = 6C6 = 1
Probability of matching all 6 numbers = 1/3838380

b) Number of ways to match the 5 numbers = 6C5
Number of ways to select a non matching number = 34C1
Probability of matching 5 numbers = (6C5 * 34C1)/3838380 = 204/3838380 = 17/319865

c) Number of ways to match the 4 numbers = 6C4
Number of ways to select 2 non matching number = 34C2
Probability of matching 4 numbers = (6C4 * 34C2)/3838380 = 561/255892

d) Since the probability of matching all 6 numbers is 1/3838380, it would take 3838380 weeks to match all 6 numbers once.

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