Suggesting a pilot has to do 25 combat sorties, and an average sorty has a fatality rate of 4% (i.e. the chance of you getting killed in a single sorty is 4%)What is the chance of you surviving the 25 sorties?
The chance of living through a sorty is 96%, so the overall chance of surviving is .96^25 or about 36% according to my calculator.
His chance of surviving a single sortie, given a 4% fatality rate, is 96%. Simply raise that to the 25th power (i.e. he has to survive every single mission, thus .96 x .96 x .96…etc.), and his chances are 36%.
Looks like Lumpy can type faster than me.
Assuming that each sortie is an independent event then
4% chance of fatality on sortie #1 = 96% chance of survival of sortie #1.
Chance of surviving sortie #1 & #2 = .96^2
and so on…
Chance of surviving sorties #1- #25 inclusive = .96^25 = 0.360396717 = 36%
The answers above are correct assuming that surviving the sorties are independent random variables and that the chance of surviving any given sorty is exactly 4%, but those are probably not reasonable assumptions. There’s going to be variance in the survival rate, and pilots who’ve flown 10+ sorties probably have better odds of surviving the next one than pilots who have never flown at all. 0.96[sup]25[/sup] is a reasonable first estimate, but very likely not quite right.
Key point.
BTW, I don’t think it’s a coincidence that OP picked numbers (4%, 25) that multiply to produce 1. As the numbers become more extreme (say, 0.1%, 1000), the answer approaches the reciprocal of Euler’s constant 36.7% = 1/2.71828 (This is good to remember when no calculater is handy and “ballpark” is good enough.)
Exactly. As my Dad the Infantry Combat veteran pointed out, the “new guy” had a much greater chance of being killed. After a certain amount of time in the front lines, you delevoped a sixth sense.
The same has been said by pilots, that surviving your first dogfight was critical.
Absolutely. If you don’t survive your first dogfight, you’ll never make it through the next one.
Da winnah! And new champeen!
But let’s say the chances of surviving all 25 sorties is 36 per cent. Why isn’t this number reset to 50 per cent every time the pilot returns, whether more experience is gained or not, as it is every time a coin is flipped?
Any percentage that could be attached due to experience is subjective.
There are these things called parachutes…
(which is why GB won the Battle of Britain, or at least a major reason)
Where does 50% come in, unless you’re saying the pilot either survives or he doesn’t. The OP specifies that the odds per sorty are 96/04.
Assuming we don’t allow for a pilot to gain experience, it is “reset” to 96% for every sortie. The only difference is that we’re weighting the probability of survival at 96% instead of 50%.
When we have multiple events that have to occur together, intuitively, two events happening is a less likely occurrence than one or the other. Surviving two sorties requires you to be lucky twice. So, even though the probability stays the same, the result isn’t still 96/4 at the end, since surviving 25 sorties is less likely than surviving 1. It’s true that surviving the 25th sortie is still .96; the smaller number recognizes the difficulty of getting there by surviving the first 24.
For example, if we asked the same question - if we flip a coin and keep going until we get a Tail - what’s the probability of “surviving” 25 flips? It would be 50% per flip, or .50^25, which is about .00000003.
The number IS reset. Sort of. Probability changes as the experiment progresses. After the pilot returns from mission #1, the probability of surviving mission #1 is no longer 0.96. It is now 1.00. So now the probability of surviving all 50 missions is 1.00 * 0.96^49. Similarly, after he returns from mission #2, it changes to 1.00^2 * 0.96^48. And so on.
And, of course, if he dies on any mission, the probability of surviving that mission is 0.00. Since anything times zero is zero, the probability of surviving all 50 missions is also zero, regardless of which mission it is he dies on.
The question as originally written applies only when we don’t know the outcome of any missions.
I see. Thanks.
I haven’t done the math. But I have a feeling the answer is around 36%.
Freeman Dyson worked for the British airforce during the war, and discovered (very disturbingly) that while early in the war, experienced pilots did have a better chance of survival in their later sorties, by later in the war, there was no “experience” effect - the odds of survival were the same for the new guy and the experienced guy alike.
This is where I thought the OP was heading. A misunderstanding of statistics (which many people have) would lead someone to incorrectly say, “Well, if you have a 4% chance of dying and you do it 25 times, then 4 X 25=100% chance of dying!!!”
Sorry for resurrecting this thread but what was the reason for this?
Thanks
Dyson discusses his work for Bomber command in this (very interesting) article http://www.technologyreview.com/article/406789/a-failure-of-intelligence/4/ (I’m linking to page 4 because it’s most relevant). Reading between the lines, it seems like the bombers had no real defense against German fighters, and thus survival rate depended solely on the German’s success rate at hitting their targets - so the experience of the bomber crew was irrelevant.
That’s a fascinating read. Page three ties in with the OP:
“Our bomber losses were rising, close to 4 percent for the third attack and a little over 4 percent for the fourth. The Germans had learnt quickly how to deal with WINDOW. Since they could no longer track individual bombers with radar, they guided their fighters into the bomber stream and let them find their own targets. Within a month, loss rates were back at the 5 percent level, and WINDOW was no longer saving lives.”
For fighters I surmise that, by 1944, the Luftwaffe was on its last legs, and the big enemy was flak, which was indiscriminate and shot down experienced and novice pilots alike. In which case the deciding factor would have been mission planning and intelligence.