In the .... Lotto, you may pick six different numbers from theset {1, 2, 3, ....

Question

In the .... Lotto, you may pick six different numbers from theset {1, 2, 3, ......, 51, 52}. If all six of your numbers arerandomly selected from 52, you share in a grand prize pool. If theydraw five (four) of your six numbers, you share in the second(third) prize pool.
A) What is the probability that you will share in the firstprize pool? B) What is the probability that you will share in the secondor in the third prize pool?
In the .... Lotto, you may pick six different numbers from theset {1, 2, 3, ......, 51, 52}. If all six of your numbers arerandomly selected from 52, you share in a grand prize pool. If theydraw five (four) of your six numbers, you share in the second(third) prize pool.
A) What is the probability that you will share in the firstprize pool? B) What is the probability that you will share in the secondor in the third prize pool?

Explanation / Answer

You share in first prize when they draw all 6 numbers that youhave selected.
P(they draw all 6 numbers that you have selected.)
= # of ways they draw 6 numbers that you have chosen/Total number of ways of drawing 6 numbers
= 1/ 52C6 = 4.912 * 10 -8
P(second prize) = P(they draw 5 numbers that you have chosenand 1 number other than the 6 numbers that you have choosen)
= # of ways they draw 5 numbers that you have chosen and1 number other than the 6 numbers that you have chosen/Total numberof ways of drawing 6 numbers
# of ways they draw 5 numbers that you have chosenand 1 number other than the 6 numbers that you havechosen = # of ways of choosing the 5 numbers * # of ways ofdrawing the one number from (52-6) = 46 numbers
= 6C5* 46C1
therefore P(second prize) = 6C5*46C1/52C6 = 1.356 *10-5
P(third prize) = P(they draw 4 numbers that you havechosen and 2 numbers other than the 6 numbers that you havechoosen)
= # of ways they draw 3 numbers that you have chosen and 2numbers other than the 6 numbers that you have chosen/Total numberof ways of drawing 6 numbers
# of ways they draw 4 numbers that you have chosen and 2number other than the 6 numbers that you havechosen = # of ways of choosing the 5 numbers * # of ways ofdrawing the one number from (52-6) = 46 numbers
= 6C4* 46C2
therefore P(Third prize) = 6C4*46C2/52C6 = 0.0007626
Hope this helps you. Feel free to ask for anyclarifications.

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