100% chance of success?

本站添加: 2026-05-28

[00:00:00]Well, that is not how probability works.

[00:00:02]But the people in the comments were ruthless. So many jokes were made about this without actually explaining why she was wrong. Many people commented the right answer, but without actually explaining how the probability works. So let's do that. One quick way to see that her argument didn't make sense is using the example of a fair coin. Suppose that we succeed when we get heads and the probability of heads is a half. then it is not enough to just toss the coin twice because there is the possibility that we get two tails and we lose. So even though the probability is a half after two coin tosses we still have that the probability of succeeding is only three4s. If you toss the coin three times it is still not guaranteed that one of them will be heads because there is a possibility of three tails in a row. And if you count all the possibilities, there are eight possibilities in total. Seven of them are favorable because there's at least one heads. And therefore, the probability that we succeed in three coin tosses is 7 over 8. But you can see that even in this case, counting the favorable cases is starting to get complicated. So that's why in probability many times it's advantageous to count the not favorable cases because many times those are easier to count.

[00:01:24]For example, in here there's only one non-favorable case which is three tails.

[00:01:29]For example, in the case of two coin tosses, we can come up with that 3/4s we computed before by computing one minus the probability of the bad cases. So the probability of two tails, which is 1 minus a half * a half because the probability of two tails is that we get tails and we get tails. So it's a half times a half. So the probability of at least one heads is 1 minus a fourth which is 3/4s. Similarly, if we have three coin tosses, then we can compute the probability of at least one is heads by computing one minus the probability of all of them are tails. So it's 1 minus a half * a half time a half. So it's 1 minus an eighth. So the probability that we were looking for is 7/8s. And if you want to know what is the probability that you get at least one head in n coin tosses that will be one minus the probability of n tails in a row. So it will be 1 minus a half* a half* a half. So 1 - a half to the nth power. Now we can model the case in her message as an unfair coin where heads or succeeding is one in 100 and tails is not succeeding. So 99 in 100. And we want to know what's the probability that we get at least one heads in n tosses.

[00:02:48]It's again advantageous to compute this probability as one minus the probability of not succeeding. So of n tails in a row. And the probability of n tails in a row is the probability of the first one is tails. 99 over 100. The second one is tails. The third one is tails up to the nth one is tails. So it's 1 minus 99 over 100 to the nth power. And now we can make a table of values of this probability for some values of n. For example, when we do two attempts, we are at about 2% chance of succeeding. With five attempts, we are about 5% chance of succeeding. With 10, 9.5%. With 50 39%.

[00:03:27]100 times we are not at 100% chance of succeeding. We are still only at 63.3%.

[00:03:35]But by 200, we are at 86%. And by 500, somewhere between 460 and 500, we are at 99% chance of success.