Wikipedia:Reference desk/Archives/Mathematics/2008 December 18
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December 18
[edit]Simple probability
[edit]If an event has a probability p of producing a certain result, and n events are observed, what is the proabability that the result happens f times. This seems really easy, but I can't get it! I can work out the probability that it happens at least once, but I don't think that's relevent. Thanks. —Preceding unsigned comment added by 70.52.47.6 (talk) 04:32, 18 December 2008 (UTC)
- A few hints: (1) In a sequence of events, what is the probability that the result in question is observed in exactly the first events, but not the other ? (2) Consider, more generally, any particular sequence of events in which the result is observed times. What is the probability that that particular sequence of events is observed? (3) In how many ways can you have exactly occurrences of the result distributed among events? --173.49.12.32 (talk) 05:23, 18 December 2008 (UTC)
- Okay, unless I misunderstood you're hints/questions, the answers are 1: p^f, 2:(f!(n-f)!)/n!, 3: n!/(f!(n-f)!). Seeing as I'm not getting anywhere with this, I'll assume I misinterpreted your questions. Sorry. —Preceding unsigned comment added by 70.52.47.6 (talk) 05:41, 18 December 2008 (UTC)
- Try making a probability tree - Mgm|(talk) 12:53, 18 December 2008 (UTC)
- Also see binomial distribution. -- Jao (talk) 12:57, 18 December 2008 (UTC)
- If there are f occurrences in n events, there must be n-f non-occurrences. The probability of an individual non-occurrence must be 1-p, so the first answer p^f is wrong, it should be p^f.(1-p)^(n-f) As that is the probability of one particular way of getting f occurrences in n events, you need to know how many different ways there are.→81.132.236.144 (talk) 13:56, 18 December 2008 (UTC)
- Note that this is only true for n independent events. Imagine a coin that always alternates between head and tail. Each single toss has a probability of 50% to be heads, but a million-toss sequence will always contain *exactly* half a million heads. Taken literally, the original question is unanswerable. —Preceding unsigned comment added by 84.187.93.95 (talk) 01:39, 19 December 2008 (UTC)
- You're mistaken. It's not ONLY if they're independent. It's what happens IF they're independent, but not ONLY if. For example, what if I toss a dime and quarter that are so entwined that if the dime comes up "tails" then so does the quarter, and that happens 1/4 of the time, and if the quarter comes up "heads" then so does the dime, and that happens 1/4 of the time. The other 1/2 of the time consists of the dime showing "heads" and the quarter showing "tails". Thus the numbers of "heads" that appear is either 0, ,1, or 2, with respective probabilities 1/4, 1/2, ,1/4—the same probabilities as if they were independent. Michael Hardy (talk) 01:58, 19 December 2008 (UTC)
- They are in fact independent. Your formulation of the problem does not change that, the criterion for independence is the conditional probability.
- No, they are not. We have P(dime shows tails and quarter shows heads)=0!=1/16=P(dime shows tails)P(quarter shows heads). Algebraist 01:03, 20 December 2008 (UTC)
- They are in fact independent. Your formulation of the problem does not change that, the criterion for independence is the conditional probability.
- You're mistaken. It's not ONLY if they're independent. It's what happens IF they're independent, but not ONLY if. For example, what if I toss a dime and quarter that are so entwined that if the dime comes up "tails" then so does the quarter, and that happens 1/4 of the time, and if the quarter comes up "heads" then so does the dime, and that happens 1/4 of the time. The other 1/2 of the time consists of the dime showing "heads" and the quarter showing "tails". Thus the numbers of "heads" that appear is either 0, ,1, or 2, with respective probabilities 1/4, 1/2, ,1/4—the same probabilities as if they were independent. Michael Hardy (talk) 01:58, 19 December 2008 (UTC)
The way the word "event" is used above conflicts with standard usage in the field. If I throw a die, getting an outcome of 5 or more is an event; getting an outcome of 4 or less is the complementary event. Throwing the die is NOT an "event"; sometimes it might be called an "experiment" or a "trial".
At any rate, binomial distribution is the article that addresses the question posed here (assuming the trials are independent). Michael Hardy (talk) 01:52, 19 December 2008 (UTC)
- why when talking about Probability everybody is so definitely certain to be right? :) PMajer (talk) —Preceding unsigned comment added by 84.221.198.219 (talk) 09:16, 21 December 2008 (UTC)