Should you buy all 175.2 million Powerball combinations?

Update 1: The Powerball annuity jackpot has been increased to $500 million, and the cash value is now $327 million. (Tuesday, Nov 27 3:31 PM)
 
With the Powerball jackpot now at $425 million, it is logical for players to wonder if they should buy tickets for all the possible combinations. At first glance, one might think this would be a winning strategy for picking up an easy $74.6 million. 

However, before one recklessly spends this much money, there are several factors that that must be considered.

1. In the 2012 Powerball game, each ticket costs $2. Since there are approximately 175.2 million combinations, one would have to spend $350.4 million to accomplish this.
2. The $425 million jackpot is the annuity value which is paid in 30 varying installments (29 years). This implies that the breakeven 0 profit financing rate is 1.144%. So, if one borrowed the $350.4 million, the borrowing rate must be less than this interest rate in order to make a profit. If the borrowing rate is more than 1.144%, you will lose money.
3. The cash value being offered today is $278.3 million. If you take the cash, you will immediately lose $72.1 million.
4. In order to guarantee yourself a winning ticket, you must buy all the 175.2 unique combinations. If you are lazy and buy 175.2 quick picks, then there is a 36.8% chance that you will not have the winning combination.
5. When the jackpot is this high, we can expect that approximately 120 million other tickets will be sold. Of these, there will be a 50% chance that one of these will be a winning ticket. This means that there is a 50% chance that you will have to share the jackpot prize with at least 1 other winner. If this happens, you might win only $212.5 million annuity or $139.2 million cash or less. Again, you will take a loss.

As you can quickly see, there is a very high probability that even if you buy all the unique combinations, you will end up losing millions of dollars if you purchase all the combinations. Thus, our advice is:

No, You should NOT but all the Powerball combinations.

But, what if no one wins the jackpot in the next drawing and the Powerball grows to $600 million. Should you buy the combinations then?
To help you answer this, we present the two graphs below. The first shows the chances of there being a losing Powerball as ticket sales grow to 500 million. As you can see, the chances decrease as sales increase. When 200 million tickets are sold. there is a 30% chance that nobody will win. When ticket sales hit 500 million, there is less than a 6% chance that there will be no winner.

The second graph shows the probabilities of having more than one winner. At sales of 200 million, there is a 40% chance of multiple winners. When sales reach 500 million, there is a 67% chance that there will be more than one winning ticket.

So, even as the temptation to purchase all combinations increases as the jackpot value grows, the likelihood that the prize will be shared by more than one winner also increases. So, even if the annuity jackpot reached $1 trillion, it would most likely be shared by 2 or more winners, which means that each winner would only receive a jackpot prize equal to or less than the $425 million currently offered.

Thus, we continue to recommend that you should:

NEVER buy all the Powerball combinations!