Question

In the Powerball® lottery, the player chooses five numbers from a set of 69 numbers without replacement and one “Powerball” number from a set 26 numbers. The five regular numbers are always displayed and read off in ascending order, so order does not matter. A player wins the jackpot if all six of the player’s numbers match the six winning numbers.

a. How many different possible ways are there to select the six numbers?

b. How many tickets would someone have to purchase to have at least a one in a million chance of winning the jackpot?

c. What is the probability that none of the player’s numbers match any of the six winning numbers? (Assume this player only has one ticket.)

d. The player wins the second prize if the player matches the first five numbers but does not match the Powerball. The player wins third prize if the player matches the Powerball but only matches four of the other five numbers. What is the probability that a player who buys a single ticket will win at least third prize?

Answer #1

a) number of ways =N(select 5 umber of 69 and then select 1
number from
26)=^{69}C_{5}*^{26}C_{1}
=11238513*26=292201338

b)

number of tickets to purchase =292201338/10^{6} =292.20~
293

c)

probability that none of the player’s numbers match any of the
six winning numbers
=^{64}C_{5}*^{25}c_{1}/(^{69}C_{5}*^{26}C_{1})

=0.652334

d)

probability that a player who buys a single ticket will win at least third prize

=P(win powerball)+P(second prize)+P(third prize)

=^{5}C_{5}*^{1}c_{1}/(^{69}C_{5}*^{26}C_{1})
+^{5}C_{5}*^{25}c_{1}/(^{69}C_{5}*^{26}C_{1})
+^{64}C_{1}*^{5}C_{4}*^{1}c_{1}/(^{69}C_{5}*^{26}C_{1})
=1/292201338+25/292201338+320/292201338

=1.18412*10^{-6}

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