Read this.

As many of you know, IHG just launched a sweepstakes that stacks on top of their Accelerate promotion. The sweepstakes offers 1,000 points after your first stay, and then a mystery prize for each subsequent stay. There have been plenty of posts on the promo, so I won’t go into detail about the prizes, but for more information, you can read here.

Now, because its a sweepstakes, IHG is legally required to offer a method of entry that doesn’t require a purchase (otherwise it would be considered gambling). Deals We Like scoured the fine print and found that this method of entry involved—surprise, surprise—writing postcards*. 94 of them in fact. And under the assumption that you win at least 500 points per entry, you’d come out with 47,000 IHG points after spending $46.06 in postage and a lot of time writing. Not bad.

But wait. My friend Shane pointed out to me that odds are not the same thing as probability. Probability is a fraction of successes to total possible outcomes, whereas odds are a fraction of successes to failures. This is a subtle difference but makes what otherwise seems like a sure thing — 1:1.18 odds for the lowest prize of 500 points — not 85% likely (1/1.18) but 46% likely (1/(1+1.18)).

Where does that leave us? Well, as I explained in a post a couple of months back as to whether or not the SPG Open the World promo was worth your time, you need to calculate the expected value of your entries to determine whether or not they are worth your time. As before, I’m going to exclude the grand prizes since the probability of winning is so small as to be negligible.

In that case, given the following odds (reproduced from the official terms):

Points Prize Odds
5000 1:42.86
2000 1:28.57
1000 1:14.29
500 1:1.18

Your expected value per postcard is:

EV = 5000 * 1/43.86 + 2000 * 1/29.57 + 1000 * 1/15.29 + 500 * 1/2.18

476.4 points. Incidentally, this isn’t too far off from the stated assumption of 500 points per play.

However, given that the payout isn’t guaranteed, we’re better off looking at the range of possible payouts and their probabilities. Mathematically, this can be summarized by the standard deviation, which describes how far off most results are from the average. A lower standard deviation corresponds to more of a ‘sure thing’ and therefore something that is low-risk, whereas a higher standard deviation corresponds to a very risky proposition.

For our odds above, the standard deviation is a whopping 811 (for how to compute the value, see this handy explanation), which is huge given that it’s twice the average payout. The problem, therefore, is that this doesn’t give us a real intuition (or practical information for that matter) about how well we’re expected to perform over 94 postcards.

Thankfully, 94 postcards is enough for the law of large numbers to start to come into play (the law of large numbers basically states there will be less variation in our payoffs over many plays than a single one). I’ll spare you the math (given that there are research papers about it since it’s a hard problem), but my friend Nick was kind enough to write a program to compute the aggregate outcomes over 94 entries**. I’ve graphed them below (you can find a link to an interactive version here; I’ve also included the generated data and the program to compute the probabilities here and here respectively):

IHG Promo Payoff Distribution

Basically, you can expect to win between 30,000 and 50,000 points with reasonable probability. Not a sure thing by any means, but if you value 30,000 points at more than the amount of time it will take you to crank out 94 letters, then by all means, go for it.

As for me, I’m actually going to participate, because it makes for a good story, and I enjoy doing ridiculous things***, but it’s not necessarily as good a deal as it looks at first brush.

* As tokyohyattfan points out, technically you have to put a 3×5 card inside an envelope.

** We ignored the 5 prize cap on the 5,000, 2,000, and 1,000 point prizes to simplify the analysis, although it lowers the expected payout by a bit (i.e. shifts the graph left).

*** Case in point:

Michael and I take over a desk for our IHG letter-writing campaign.

Michael and I take over a desk for our IHG letter-writing campaign.