What we can learn from the math behind small probabilities

In high school stats class we had a homework assignment during our section on probabilities that went something like this:

A rock climber estimates his odds of dying on an individual climb are 1 in 1,000. What is the probability that he’ll die before his 1,000th climb?

Kind of morbid, but I’ve always remembered it because it has a non-intuitive answer for reasons that we can apply to a lot of real-world situations.

Many people will answer that the climber’s odds of dying after 1,000 climbs are 1 in 1,000, but that’s incorrect. It’s true that his odds of dying on an individual climb are 1 in 1,000, but we have to account for the fact that he’s doing those individual climbs 1,000 times.

The math works out like this:

His odds of dying on an individual climb are 1 in 1,000 (0.001) meaning his odds of surviving are 999 in 1,000 (0.999). His odds of surviving 2 climbs is 0.999 * 0.999, his odds of surviving 3 climbs is 0.999 * 0.999 * 0.999, his odds of surviving n climbs is 0.999^n. For 1,000 climbs, his probability of surviving is 0.999^1000 or 0.368, meaning he’ll die 63.2% of the time before he reaches his 1,000th climb. Ouch.

How can we us this to our advantage? Consider a more uplifting example:

You’re a junior developer and estimate there’s only a 1 in 50 chance of getting hired by a large Silicon Valley software company. What are your odds of getting hired after 10 interviews? 1 – 0.98^10 = 18%. After 25 interviews it’s 40%, after 50 interviews it’s 64%. For each individual interview there’s a 1 in 50 chance of getting hired, but because you keep interviewing, that 1 in 50 will likely eventually happen [1].

To sum it up:

Unlikely risks will likely eventually occur if you’re exposed to them often. Similarly, unlikely opportunities will also likely eventually occur if you’re exposed to them often.

Use the latter to your advantage by pursuing big opportunities even if you don’t think it’s likely they’ll ever happen because if you keep at it, chances are one eventually will.

[1] The reality in both situations is a little bit more complicated because the odds aren’t static. A rock climber that begins climbing with 1 in 1,000 odds of dying will improve his skills as he climbs which decreases his odds of dying. But he or she will also likely be increasing the difficulty of the climbs, possibly negating the decreased odds of dying due to the improved skill. Similarly, the junior developer will hopefully be improving his interviewing skills along the way, making his odds of getting hired better than 1 in 50 the more he interviews.

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