Tricky probability question

stubby5000

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I've spent quite a bit of time noodling with this one, and I can't come up with a solution. Here's the scenario:
Take 30 seconds and divide it up into 10 millisecond periods (there are 3000 of them in 30 seconds). In each of these periods, there are two possible outcomes (filled or not filled) with are NOT equal in likelihood. The likelihood of a period being filled is low (let's say it's a 0.01% chance).
I want to come up with a formula that assesses the likelihood of certain events occurring during the whole 30 seconds. Examples include things like having two filled periods in a row, or three filled periods in a row, or two filled periods out of five, or five filled periods out of twelve, etc. When I started this I initially assumed since each period has a 0.01% chance of being filled, and there are 3000 total periods or "chances" to fill them, 0.01 * 3000 = 30% chance of one filled period in 30 seconds. I then assumed that the chances of two in a row was (0.01*0.01)*(3000/2)<---the number of chances you have to get this result. This equaled 0.15%, which seemed reasonable, until I started considering outcomes with empty periods in them (e.g., two filled periods out of five). I assumed that the likelihood of the two filled periods was still 0.01*0.01, but now you only had 3000/5 chances to get it. This gave a result of 0.06%, which of course doesn't make sense as two out of five should be more likely than two in a row.
I've tried watching Khan Academy videos on probability but they just deal with simple stuff like coin flips.
I would very much appreciate any help you could provide on this!
 
I've spent quite a bit of time noodling with this one, and I can't come up with a solution. Here's the scenario:
Take 30 seconds and divide it up into 10 millisecond periods (there are 3000 of them in 30 seconds). In each of these periods, there are two possible outcomes (filled or not filled) with are NOT equal in likelihood. The likelihood of a period being filled is low (let's say it's a 0.01% chance).
I want to come up with a formula that assesses the likelihood of certain events occurring during the whole 30 seconds. Examples include things like having two filled periods in a row, or three filled periods in a row, or two filled periods out of five, or five filled periods out of twelve, etc. When I started this I initially assumed since each period has a 0.01% chance of being filled, and there are 3000 total periods or "chances" to fill them, 0.01 * 3000 = 30% chance of one filled period in 30 seconds. I then assumed that the chances of two in a row was (0.01*0.01)*(3000/2)<---the number of chances you have to get this result. This equaled 0.15%, which seemed reasonable, until I started considering outcomes with empty periods in them (e.g., two filled periods out of five). I assumed that the likelihood of the two filled periods was still 0.01*0.01, but now you only had 3000/5 chances to get it. This gave a result of 0.06%, which of course doesn't make sense as two out of five should be more likely than two in a row.
I've tried watching Khan Academy videos on probability but they just deal with simple stuff like coin flips.
I would very much appreciate any help you could provide on this!

Have you learned about the binomial probability distribution? That is what you need here (if you aren't looking for a given number in a row, but just any out of the 3000).

But you also have to think carefully about whether you want, say, the probability of exactly 2 out of 5, or at least 2 out of 5 (which is often more relevant to whatever your purpose is).

To take a simple case, the probability of at least one of 3000 slots being filled is 1 minus the probability that none will be filled, which comes to 1 - 0.99^3000 = 0.99999999999991954, not 30% as you suggested. This will almost always happen. If you meant exactly 1 (so that having 2 or more filled doesn't count), then there are 3000 successful outcomes (each of the slots that could be filled), each with probability 0.01*0.99^2999, for a total of 3000*0.01*0.99^2999 = 0.0000000000024382.

I haven't had to use binomial probabilities here, though that last calculation came close; you will need it for other cases. You should be able to find it easily with a search (including Khan).
 
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