(Also, please don’t take this as mocking or disrespectful — I just have a rather mathematical mind, that’s all, and when I hear a suspicious-sounding number I have to do my best to verify it.)

In watching the video documenting the second crash, the US ambassador mentioned that, for a plane of 125 people of which 50 survive, the odds of a husband and wife surviving are 1 in 16. I’m not so sure of that probability, but I’d like someone to check my math.

Given that we know 50 people survive (unstated in the ambassador’s description, but a reasonable assumption given that if all passengers survive this probability would be 100%, not the cited number), the probability of the husband-wife team surviving is the number of possible sets of survivors which include both husband and wife, divided by the number of possible sets of survivors. Thus we have:

# sets of survivors including husband/wife = 123 choose 48
# sets of survivors = 125 choose 50

In the latter case, we simply choose 48 survivors from the remaining pool of 123 passengers. After expanding the combinations into factorials, this then gives us a figure of:

P(couple survive) = (123! / (48! * (123 – 48)!)) / (125! / (50! * (125 – 50)!))
= (50 * 49) / (125*124)
= 49 / 310
= 0.158064516

This number, if it’s correct, isn’t the 1 in 16 the ambassador mentioned; it’s actually a bit better.

There’s one other possibility for the probability of 1 in 16: it’s the probability of a couple surviving given that any number of passengers survive, from 0 to 125. In that case we must generalize the above probability to allow 48 to vary from 2 to 125 (can’t include 0 or 1 because a couple can’t survive in either case, and the formula is degenerate), sum all those values, and divide by the number of possible survivor counts. Let P(n) be the probability of a couple on a 125-passenger plane surviving a plane crash where 2<=n<=125 people survive. The general formula in this situation is:

P(n) = (125 – 2) choose (n – 2) / 125 choose n

…and the total probability, assuming equal probability of all numbers of survivors, is then:

P(couple survive) = (sum of P(n) for n from 2 to 125 inclusive) / 126

Simplifying P(n) we have:

P(n) = 123! / (n-2)! / (125! / n!)
P(n) = 123! * n! / ((n-2)! * 125!)
P(n) = 123! * n * (n-1) / 125!
P(n) = n * (n-1) / (125*124)

We can then remove the denominator and divide the total of all probabilities by that value. The final result, then, seems to be 1 in 3, as seemingly demonstrated by the following JavaScript calculation (presumably modulo floating-point error in the last digit; it wouldn’t be hard to sum n**2 and -2n across all values, to be sure, but I’m feeling lazy after crunching on this for awhile):

javascript: var ps = 0; for (var n = 2; n <= 125; n++) ps += (Math.pow(n,2) - n); alert(ps/126/(125*124))

Here’s a <a href=”javascript: var ps = 0; for (var n = 2; n link to calculate that, if comments allow HTML and JavaScript URLs through.

I’m almost depressingly rusty in performing this calculation despite having studied probability within the last year, so any help others could give by pointing out error or verifying my analysis would be greatly appreciated. Thanks! (And, one last time, my apologies if this comment offends anyone with its apparent lack of sympathy or with its relative tangency to the topic under discussion.)