Probability of 2 events happening in 5 occurances in a specific sequence?

If you have n events, and probability p of the event occurring, then the probability of x occurrences, if there is no particular order of occurrence, is





[(n!)/(x!)(n-x)!] [(p)^x][(1-p)^(n-x)]





It's called a binomial distribution.





what would be the formula if there is a particular order of occurrence, i.e. event 1 happening on the first occurence and event 2 happening on the 5th occurence

Probability of 2 events happening in 5 occurances in a specific sequence?
You would be looking at the sum of geometric random variables. One for each event.





Let X be the number of trials until the first success. X is a sum of Bernoulli trials in a way similar to a Binomial. The difference here is that for a binomial you are looking at the number of success in n trials. The Geometric is looking for the number of trials before the first success.





X has the Geometric Distribution with success probability p then:





X ~ Geometric(p)





P(X = x) = p * (1 - p) ^ (x - 1) for x = 1, 2, 3, 4, ....


P(X = x) = 0 otherwise.





As you can see, the probability mass function is derived by looking at having x - 1 failures and then 1 success.





The Expectation or Mean of the Geometric, i.e., how many trails you expect before the first success is 1/p


The Variance of the Geometric is (1 - p) / p^2
Reply:If you have n events, and probability p of an event being successful, then the probability of x successes, if there IS a particular order of occurrence, is





[(p)^x][(1-p)^(n-x)]





It's the same thing, without the part at the beginning.