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breckenridgeback t1_jaexili wrote

Suppose we have a 1% chance that something happens.

The chance it doesn't happen the first time is 99%.

The chance it doesn't happen the second time is 99%, independently of the first (we do need some degree of independence here, or your statement isn't true). So the chance it didn't happen either of the first two times is 99% times 99%, or 0.99 times 0.99, or 0.9801.

The chance it doesn't happen the third time, still 99%. So the chance it didn't happen any of the first three times is now 0.99 times 0.99 times 0.99.

You can hopefully see a pattern. The chance for the event to fail every one of the first n times is 0.99 times 0.99 times ... times 0.99, n times. In other words, that chance is 0.99^(n). And the limit of 0.99^n as n goes to infinity is 0, meaning that the probability that you manage to dodge your low probability outcome gets closer and closer to zero the larger the number of attempts you have.

Nothing about this depended on the exact 1% / 99% probabilities. All we needed was for that 99% to be less than 100%, i.e., for the 1% to be greater than 0%. As long as that's true, given some non-zero probability of success p, (1-p)^n always goes to 0 as n -> infinity (and therefore the chance you fail every time goes to zero as well).

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sdrowkcabdelleps t1_jaeylbo wrote

This guy gets it. Infinite series is a powerful tool.

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fmaz008 t1_jaf1mbt wrote

Lets just be happy it's not a serie owned by Netflix other wise they would have cancelled it already

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Holiday_Chemistry_72 OP t1_jaf0sik wrote

I wish someone can eli5 this answer, English isn't my language

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Chaotic_Lemming t1_jaf2ud5 wrote

Imagine you have a dice with 1 million sides. Your chance of rolling 123,456 is 1 in 1 million, or 0.0001%. You start rolling the dice once a day. Each day the chance the dice will land on 123,456 is 1 in 1 million. That doesn't change day to day. Now lets fast forward in time 10 million days. You've rolled the dice 10 million times. Each individual roll was still a super tiny chance. But the odds that you won't have landed on 123,456 in those 10 million roles is approximately 0.0045%. (0.999999^10,000,000)

There is still a chance that you won't have rolled a 123,456 but it is very small.

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