If the only thing you are interested in is the probability (odds) of winning the Powerball Jackpot, the Multi-State Lottery gives a concise table at their web site. We will give the same information here, but also show you how these odds are calculated.

Game Rules
   The numbers picked for the prizes consist of 5 white balls picked at random from a drum that holds 55 balls numbered from 1 to 55. The Powerball number is a single ball that is picked from a second drum that has 42 numbers ranging from 1 to 42. If the results of these random number selections match one of the winning combinations on your lottery ticket, then you win something. You can also buy a “Power Play Multiplier” option – the size of which is determined by a random spin of a “Multiplier Wheel” that has four 2’s, four 3’s, four 4’s, and four 5’s.

   In any combinatorics problem where all possible outcomes are equally likely, the probability of a successful outcome is determined by finding the number of successful combinations, and then dividing by the total number of all combinations. There are nine possible configurations that will win something in the Powerball Lottery. For each of these, the probability of winning equals the number of winning combinations for that particular configuration divided by the total number of ways the Powerball numbers can be picked.


Powerball Total Combinations
   Since the total number of combinations for Powerball numbers is used in all the calculations, we will calculate it first. The number of ways 5 numbers can be randomly selected from a field of 55 is: COMBIN(55,5) = 3,478,761. (See the math notation page or Help in Microsoft's Excel for more information on “COMBIN”).

   For each of these 3,478,761 combinations there are COMBIN(42,1) = 42 different ways to pick the Powerball number. The total number of ways to pick the 6 numbers is the product of these. Thus, the total number of equally likely Powerball combinations is 3,478,761 x 42 = 146,107,962. We will use this number for each of the following calculations.


Jackpot probability/odds (Payout varies)
The number of ways the 5 numbers on your lottery ticket can match the 5 white balls is COMBIN(5,5) = 1. The number of ways your Powerball number can match the single Powerball number is: COMBIN(1,1) = 1. The product of these is the number of ways you can win the Jackpot:  COMBIN(5,5) x COMBIN(1,1) = 1. The probability of success is thus: 1/146,107,962 = 0.00000000684425. If you express this as “One chance in ???”, you just divide “1” by the 0.00000000684425, which yields “One chance in 146,107,962”.

Match all 5 white balls but not the Powerball (Payout = $200,000)
The number of ways the 5 numbers on your lottery ticket can match the 5 white balls is COMBIN(5,5) = 1. The number of ways your Powerball number can match any of the 41 losing Powerball numbers is: COMBIN(41,1) = 41. (Pick any of the 41 losers.) Thus there are COMBIN(5,5) x COMBIN(41,1) = 41 possible combinations. The probability for winning $200,000 is thus 41/146,107,962 = 0.000000280614 or “One chance in 3,563,608.83”.

Match 4 out of 5 white balls and match the Powerball (Payout = $10,000)
The number of ways 4 of the 5 winning numbers on your lottery ticket can match the 5 white balls is COMBIN(5,4) = 5. The number of ways the losing white number on your ticket can match any of the 50 losing white numbers is COMBIN(50,1) = 50.  The number of ways your Powerball number can match the single Powerball number is: COMBIN(1,1) = 1. The product of these is the number of ways you can win this configuration:  COMBIN(5,4) x COMBIN(50,1) x COMBIN(1,1) = 250. The probability of success is thus: 250/146,107,962 = 0.00000171106 or “One chance in 584,431.85”.

Match 4 out of 5 white balls but not match the Powerball (Payout = $100)
The number of ways 4 of the 5 winning numbers on your lottery ticket can match the 5 white balls is COMBIN(5,4) = 5. The number of ways the losing white number on your ticket can match any of the 50 losing numbers is COMBIN(50,1) = 50.  The number of ways your Powerball number can miss matching the single Powerball number is: COMBIN(41,1) = 41. The product of these is the number of ways you can win this configuration:  COMBIN(5,4) x COMBIN(50,1) x COMBIN(41,1) = 10,250. The probability of success is thus: 10,250/146,107,962 = 0.0000701536 or “One chance in 14,254.44”.

Match 3 out of 5 white balls and match the Powerball (Payout = $100)
The number of ways 3 of the 5 winning numbers on your lottery ticket can match the 5 white balls is COMBIN(5,3) = 10. The number of ways the 2 losing white numbers on your ticket can match any of the 50 losing white numbers is COMBIN(50,2) = 1,225.  The number of ways your Powerball number can match the single Powerball number is: COMBIN(1,1) = 1. The product of these is the number of ways you can win this configuration:  COMBIN(5,3) x COMBIN(50,2) x COMBIN(1,1) = 12,250. The probability of success is thus: 12,250/146,107,962 = 0.0000838421 or “One chance in 11,927.18”.

Match 3 out of 5 white balls but not match the Powerball (Payout = $7)
The number of ways 3 of the 5 winning numbers on your lottery ticket can match the 5 white balls is COMBIN(5,3) = 10. The number of ways the 2 losing white numbers on your ticket can match any of the 50 losing numbers is COMBIN(50,2) = 1,225.  The number of ways your Powerball number can miss matching the single Powerball number is: COMBIN(41,1) = 41. The product of these is the number of ways you can win this configuration:  COMBIN(5,3) x COMBIN(44,2) x COMBIN(41,1) = 502,250. The probability of success is thus: 502,250/146,107,962 = 0.003437527 or “One chance in 290.91”.

Match 2 out of 5 white balls and match the Powerball (Payout = $7)
The number of ways 2 of the 5 winning numbers on your lottery ticket can match the 5 white balls is COMBIN(5,2) = 10. The number of ways the 3 losing white numbers on your ticket can match any of the 50 losing white numbers is COMBIN(50,3) = 19,600.  The number of ways your Powerball number can match the single Powerball number is: COMBIN(1,1) = 1. The product of these is the number of ways you can win this configuration:  COMBIN(5,2) x COMBIN(50,3) x COMBIN(1,1) = 196,000. The probability of success is thus: 196,000/146,107,962 = 0.001341474 or “One chance in 745.45”.

Match 1 out of 5 white balls and match the Powerball (Payout = $4)
The number of ways 1 of the 5 winning numbers on your lottery ticket can match the 5 white balls is COMBIN(5,1) = 5. The number of ways the 4 losing white numbers on your ticket can match any of the 50 losing white numbers is COMBIN(50,4) = 230,300.  The number of ways your Powerball number can match the single Powerball number is: COMBIN(1,1) = 1. The product of these is the number of ways you can win this configuration:  COMBIN(5,1) x COMBIN(50,4) x COMBIN(1,1) = 1,151,500. The probability of success is thus: 1,151,500/146,107,962 = 0.007881158 or “One chance in 126.88”.

Match 0 out of 5 white balls and match the Powerball (Payout = $3)
The number of ways 0 of the 5 winning numbers on your lottery ticket can match the 5 white balls is COMBIN(5,0) = 1. The number of ways the 5 losing white numbers on your ticket can match any of the 50 losing white numbers is COMBIN(50,5) = 2,118,760.  The number of ways your Powerball number can match the single Powerball number is: COMBIN(1,1) = 1. The product of these is the number of ways you can win this configuration:  COMBIN(5,0) x COMBIN(50,5) x COMBIN(1,1) = 2,118,760. The probability of success is thus: 2,118,760/146,107,962 = 0.014501332 or “One chance in 68.96”.

Probability of winning something
If we add all the ways you can win something we get:
1 + 41 + 250 + 10,250 + 12,250 + 502,250 + 196,000 + 1,151,500 + 2,118,760 = 3,991,302 different ways of winning something. If we divide this number by 146,107,962, we get .027317485 as a probability of winning something.  1 divided by 0.027317485 yields “One chance in 36.61” of winning something.

Corollary
   You can get a close estimate for the number of tickets that were in play for any given game by multiplying the announced number of “winners” by the above 36.61. Thus, if the lottery officials proclaim that a given lottery drawing had 2 million “winners”, then there were about 2,000,000 x 36.61 ~= 73,220,000 tickets purchased that did not win the Jackpot. Alternately, there were about 73,220,000 - 2,000,000 ~= 71,220,000 tickets that did not win anything.