In my state there are three types of ticket options for Mega Millions:
(1) A regular ticket at $2,
(2) A Megaplier ticket for $3 with enhanced non-Jackpot prizes, or
(3) "Just The Jackpot", $1 - no non-Jackpot prizes.
Which is the best "value"? To make this a tangible question, we need to define "value" in a statistical probability sense. I'm going to define this as the highest Expected Value (EV) per unit bet option. I've explained EV in previous post.
The "best" option in terms of EV per unit bet is going to be dependent on the value of the Jackpot. Just As the Jackpot goes up, the less the value of the regular and Megaplier tickets become relative to Just The Jackpot. This is just with respect to the EV (per unit bet), as the Standard Deviation of Just the Jackpot is going to be higher for the same EV.
Let's take a regular $2 ticket. Since the non-Jackpot prizes have fixed return (at least in my state, not so in California for the 5-0 2nd prize outcome, which is pari-mutuel there), we can calculate the precise EV of the non-Jackpot prizes. I've worked out that is $0.2470. For the Megaplier option, it's 3x that value.
What are the cross-over points of equal EV?
Let's define the EV of the non-Jackpot prizes as the variable "A" for the regular $2 ticket. Again, for Mega Millions, that is $2470. Furthermore, let's define the factor "k" as the reduction fraction [0,1] of the Jackpot that accounts for the pari-mutuel aspect of the Jackpot, meaning if we win, there's a chance of having to share the Jackpot with other winners. "k" will be a function of the number of tickets sold. The more tickets sold, the lower the value of "k". I've also talked about this in a previous post. Finally, let's define "p" as the probability of a single ticket winning the jackpot. For Mega Millions that is 1/302575350. In other words, given that we are a Jackpot winner, the expected value of our (pre-tax) winnings is kJ, not J. (I'm defining J here as the nominal, advertised cash value of the Jackpot).
Let's compare Just The Jackpot vs. Regular Ticket first. The value of J (our share of the cash Jackpot) where they have equal EV is as follows:
(p)(kJ) = (1/2)(p)(kJ) + A/2
The "1/2" factor on the right-hand side is to account for a $2 ticket instead of $1 for Just The Jackpot. Therefore, we can easily see that J = A / ( kp). Below that the regular ticket has the highest EV, and above Just The Jackpot.
Remember, A and k are constants given a fixed N tickets sold. "k" assumes all tickets are Quick Picks in my calculations (to follow), so we'll increase it by, say, 20% to account for people buying multiple unique combinations at a time.
What about Just The Jackpot vs. Megaplier ticket? Here, the Megaplier ticket is $3, and the non-Jackpot EV is 3A. The equation for equal EV is:
(p)(kJ) = (1/3)(p)(kJ) + A.
We can therefore conclude J = (3/2) [A / (kp)].
Next, let's compare the regular ticket with the Megaplier ticket. The equation is:
(1/2)(p)(kJ) + A/2 = (1/3)(p)(kJ) + A
Therefore, J = 3A / (kp)
What are the theoretical values of "k" - given all tickets sold are Quick Picks. Here are the numbers I've worked out for Mega Millions. "N" is the number of tickets.
| k |
N |
| 0.7401 |
200M |
| 0.6446 |
300M |
| 0.5660 |
400M |
| 0.5008 |
500M |
| 0.4465 |
600M |
| 0.4007 |
700M |
| 0.3619 |
800M |
| 0.3286 |
900M |
| 0.2998 |
1000M (1B)
|
As mentioned, to account for non-Quick Pick tickets, let's reduce these numbers further by 20%, or a factor of 0.8.
Take an example of 500M tickets sold. Adjusted "k" is therefore 0.5008 x 0.8 = 0.4006. Compare Just The Jackpot to regular $2 ticket. Therefore
J = $0.2470 x 302575350 / 0.4006 = $186560437. Since our current Jackpot cash value is higher than this, we have a higher EV / unit bet with Just The Jackpot. Again, we'll naturally have a higher Standard Deviation with Just The Jackpot, so it's fair to point that out as a detracting factor.
Do taxes change this calculation? No, but only if we assume all prize tiers are calculated at the same (presumed ~maximum) income tax rate. This is because the EV for both sides of the equation is reduced by the same (1 - tax rate) factor. Most people probably won't report a $2 net win, for example, but in theory (in the U.S) we are obligated to report it legally.
