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):

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:

## Leave a Reply

17 Comments on "Before you go off writing 94 postcards…"

Would postcards work? I thought the rules stated that the entries had to be placed in an envelope?

Yeah — you’re correct. Will make a note.

This would increase postage costs as well, possibly skewing the risk/reward further into “ah fahgedaboutit” territory.

Postage costs include a 49 cent stamp, so that part was already correct. Was just being a bit sloppy with how I was referring to the entry method.

Need to calculate the cost for the 94 pieces of envelopes too….

Where are our envelope stuffing pictures! 😀

Lol totally forgot to post. Will do that in a bit :p

I can’t tell for sure since I can barely read Java code, but from your data set, it appears that you’re significantly overestimating the expected value. The Ts&Cs state that you can only win a total of 5 (any combination) of the “Group B” prizes (1000, 2000, or 5000 points). After you win 5, you’ll only have the 46% chance of winning 500 points. Thus the maximum possible return is 5*5000 + 89*500 = 69,500. Your data set shows small, but non-zero probabilities all the way up to 445,000.

I modeled the contest using that constraint, and my expected value for 94 entries dropped to about 31,600 (336 points per entry, well below your stated 476) with a StdDev of 4100, so 95% of the total returns should fall within the range 23K-40K points and less than 0.2% of results should go higher than 44K.

Ya, you’re correct (and I had actually made a note to that effect). One comment I’ll make about your analysis is that you’re assuming the distribution is normal, which it’s not. But otherwise the “max 5 category B prizes” constraint does lower the EV a bunch.

As a side note, the EV was computed on a single trial, so the limit doesn’t apply there.

Rules also state that the entries need to be handwritten. Looking at your photo it seems as if they are printed?

Nope just the envelope!

[…] ‘Dem Flyers followed up with a similar analysis (see: Before you go off writing 94 postcards…). In this post, they came up with slightly different numbers, but the same overall […]

A month and a half later, I’m curious to hear if you have a result from your efforts yet. I am about to embark on my 94 mailings to see what my points payout will be.

Submitted 65 cards, got 4×1000 and 61×500. So two free nights!

Turns out that IHG meant probability instead of odds so everything wins.

[…] I always look forward to reading their informative, educational, and entertaining posts (except for writing 94 letters to IHG, no one has time for […]