If you went to the movies in Los Angeles this summer, you may have seen the following ad from Stand Up to Cancer, a charitable program whose telethon aired last Friday night. A clear homage to MasterCard‘s long-running Priceless campaign, this ad swaps out prices for odds, ending with the sobering fact that 1 in 2 men and 1 in 3 women will be diagnosed with some type of cancer in their lifetime.

Presumably, those cancer odds are taken from The American Cancer society, which has the relevant stats posted here. When it comes to some of the other claims in the ad, though, I couldn’t help but be skeptical.

Take the bowling claim, for instance. This ad would have you believe that your odds of bowling a perfect game are 1 in 11,500. This seems quite high, even when I consider the fact that I am not a bowling master.

Let’s try to reverse-engineer this statistic. To score a perfect game in bowling, one must bowl 12 strikes in a row. Let us suppose that your probability of bowling a strike on any given frame is some number *p*. Furthermore, let’s suppose that your performance in any frame is independent of your performance in any other frame, so that you have a probability *p* of bowling a strike each time it’s your turn. Of course, whether or not these probabilities are independent is up for debate. On the one hand, bowling many strikes in a row may make you more anxious about keeping your streak going, which may in turn decrease your probability of another strike; but on the other hand, if you are an adrenaline junkie who thrives in the limelight that only a bowling alley can provide, perhaps such a chain would make it more likely for your streak to continue. In any event, these are questions better suited to a psychologist rather than a mathematician, so for simplicity let us ignore them here.

If you have a probability *p* of bowling a strike, and a perfect game requires 12 strikes, then the probability you will score a perfect game is the product of 12 copies of *p* (one for each strike), or *p*^{12}. If the above ad is to be believed, this probability must equal 1/11500. In other words,

Taking the twelfth root of each side, we can then conclude that

In other words, your odds of bowling a perfect game are 1 in 11,500 if and only if the probability that you’ll bowl a strike is around 45.88%. This seems like an extremely generous probability to give to the population at large. After all, who among you or your circle of friends bowls a strike, on average, every other frame? Perhaps I bowl exclusively with people who are not very good (myself included), but I would think a fairer probability for the entire population would be closer to 20% or 30% (maybe even this is too generous).

What to these two alternatives yield for the odds of bowling a perfect game? Well, if *p* = .3, then *p*^{12} is approximately 1 in 1,881,676; for p = .2, the odds plummet to 1 in 244,140,625. Both of these are significantly lower than the odds cited in the ad (roughly 164 and 21,230 times lower, respectively).

It may be that the odds of witnessing a perfect game are around 11,500. For example, when you go to a bowling alley, there may be experienced players practicing. Moreover, there are many games occurring simultaneously at a bowling alley, thus increasing the odds that at least one of them will be a perfect game. But saying “you have a 1 in 11,500 chance of seeing someone else bowl a perfect game” doesn’t sound as sexy as “you have a 1 in 11,500 chance of bowling a perfect game,” I suppose.

Some of the other odds are questionable as well. For example, the National Weather Service has some data here that suggests the odds of being struck by lightning in a given year are about 1 in 500,000, not too far off from the ad’s claim of 1 in 576,000. This isn’t an apples to apples comparison, though, because the ad does not specify that these are the odds you will be struck by lightning *in a given year*. If we take into account the average lifespan in the United States (approximately 78 years), then the probability of being struck by lightning (in one’s lifetime, not in one particular year) is closer to 1 – (499,999/500,000)^{78}, which is around 1 in 6,411. Much higher, you’ll note, than the odds of bowling a perfect game. (Once again, of course, we are assuming that the odds of being struck by lightning don’t vary from year to year, and that the odds of being struck in one year are independent of the odds in any other year.)

If the point of the ad is to give us an intuitive understanding of how likely it is for us to develop cancer, then it seems important to give benchmarks that are accurate and relatable. Most people have bowled, but few people will have a good intuitive understanding of what it means to face odds that are 1 in 1.8 million (the odds of bowling a perfect game if you get a strike 30% of the time). I think the point of the ad is understood regardless, but it’s a shame that the claims leading up to this point weren’t checked more thoroughly. Indeed, many of the odds quoted in the ad can be found here, a humor website that offers no sources for any of its statistics.

Perhaps Jon Stewart was in charge of fact-checking. Given his lack of understanding about the nature of this program, this is perhaps the most reasonable explanation.

If you don’t mind my taking a stab at their bowling odds, I propose a slightly different reverse engineering. When I and my fellow mill-hunks go to the local alley, I’ve often been impressed by the number of plaques they have up commemorating perfect games. At the nearest alley, the oldest plaque is from around 2002, and the plaques span a substantial portion of the length of the alley. I’d estimate maybe around 200 perfect games are commemorated there. If 1 of every 11500 games are perfect, then would put about 2.3 million games having been bowled at that alley since 2002. A little under 800 games a day seems reasonable as an estimate at a 30-lane alley. 40% strikes still seems pretty high (although my fellow millers and I average around 76% strikes), but maybe it is reasonable after all.

Another possibility is that, as you mentioned with the lightning, it’s the probability that a random American has ever bowled a perfect game. Given that the population of the US is estimated at 310 million, that means only 27000 people in the US have ever had to bowl a perfect game to get the 1:11500 ratio. Again, seems plausible, if completely incalculable.

these both seem like good ideas. I found that statistic in a number of places on the internet, but could never find a source for it.

are you telling me you have a 76% strike rate? I will never go bowling with you.

76% Stirke rate, really? Are you saying you get 9 out of 12 strikes per game? Don’t think so.