The binomial distribution (e.g., B(n,p)) is the sum of n independent bernoulli p trials/distributions. You are throwing the dart 3 times and you want to hit twice exactly, that can happen three ways (hit 1 and 3, 1 and 2, or 2 and 3); and summing that all up can be annoying (especially when you are looking at something with more trials than this. That is where the binomial PDF (probability density function) comes in. (nCr)*p^r*q^(nr)The nCr ( =n!/(r!*(nr)!)) says how many ways can you pick r from n (in your case; how many different ways can you throw three darts and hit twice). The remainder (p^r*q^(nr)) is the probability of any one of those occurring combinations occuring; that is where the independent bernoulli trial part comes in. Since each trial is independent, the order of the events doesnt matter (e.g., p*q*p =p*p*q = q*p*p = p^2*q) . Thus, so long as you know how many possible combos there are (from the nCr) you just multiply it by the probability of an event. Given the information you have, you know p is 7/9 (the probability of success). q is usually defined as 1p, in this case 2/9. n (the number of trials) is 3, and r (the number of successes) is 2. Use the above equation and you can get your answer.Just as an FYI, your (p+q)^3 bit wont help you much since by definition p+q =1, so (p+q)^n = 1.
