One billion dollars can bring happiness, but the “how” might surprise you — The Math
Continued from this post…
[updated 1/12/16 5:53PM] Note: The original post was written when the jackpot value was $1.3B. At that point, donating led to a favorable payoff whereas keeping the (fully taxed) amount did not. The below calculations reflect this figure, but they have been re-run for the $1.5B figure, reported below. It’s not clear how many people will be playing, but one report suggests that 85% of combinations will be bought by Wednesday. I’ll leave it to you to reason whether that is informative as to how many people are playing, whether these calculations are informative, and whether it is important to consider the value to you of a three-way split — note that the risk of the three way split is considered here.
For the odds to remain in your favor, you have to have a decent shot of being the only winner. We can calculate that threshold:
O = the payoff ratio if the only winner, minus 1** = 0.70
S= the payoff ratio of sharing jackpot with one other winner, minus 1** = 0.17
p(O) = the probability of being the only winner
p(S) = the probability of being one of two winners
p(M) = the probability of being one of three or more winners
suppose M = 0, conservatively ignoring any benefit to you of being one of three or more winners
Therefore, for the odds to be favorable, it must be less than four times as likely for you to share the prize with another winner than for you to be the only winner.
In order to consider the likelihood of being the only winner, among two winners, and among three or more winners, we assign the probability, p,of picking a winning number (such as yours) to 1 in 292 million.
The following equation describes the statistical likelihood, L, of having exactly w winners besides yourself, given n randomly purchased tickets.
Comparing L derived for w=1 and w=0 and considering the threshold derived above, we find that the odds are in your favor so long as:
That is equivalent to over 178 tickets per resident of Massachusetts and over 3.9 tickets per American in Powerball states! Certainly, fewer tickets will be sold. May the odds remain in your favor!
Repeating this analysis for the lump sum suggests that the odds are still in your favor even if less dramatically, so long as fewer than 6.1*10⁸ tickets are sold (90 per MA resident or 2 per American).
Update 1/12: I repeated this analysis for the $1.5B figure. For a favorable payoff, n ≤ 8.68 * 10⁹. This is equivalent to 28 tickets per American.
** I couldn’t solve this demon, even with Mathematica, except to note that comparing these payoffs as “difference from 1” allowed for a much cleaner calculation
