Binomial Distribution and ProbabilityDate: 06/07/97 at 11:06:33 From: Bianca Clarke Subject: Binomial distribution I just don't understand binomial distribution. Can someone explain it very simply? Many thanks, Bianca Clarke Date: 06/07/97 at 15:53:26 From: Doctor Anthony Subject: Re: Binomial distribution Bianca, The binomial distribution applies when you have repeated experiments with constant probabilities of success and failure at each trial. A good example is throwing a die, say five times, and finding the probability of rolling exactly three sixes. If we use the letters S and F to represent success and failure in any trial, then one possible sequence giving the desired result would be: SSSFF The probability of this particular sequence is: (1/6)^3*(5/6)^2 Another sequence would be FFSSS and this too has probability: (1/6)^3*(5/6)^2 In fact there are many other sequences that produce 3 sixes and 2 non- sixes, and to get the total probability of 3 sixes and 2 non-sixes, we must multiply the probability of any given sequence, (1/6)^3*(5/6)^2, by the number of sequences. This in effect, is the number of different arrangements you can make with the 5 letters S and F, 3 being alike of one kind and two being alike of a second kind. In case you are unfamiliar with this problem, we shall show that the number of different arrangements is: 5! ------ 3! 2! We have 5 letters, and to start with assume they are all different letters. These letters could then be arranged in 5! ways. However three are alike of one kind and we could swap these three amongst themselves in 3! ways without giving rise to a new arrangement. Similarly two others are alike of a second kind, and these could be swapped between themselves without giving rise to a new arrangement. Our answer of 5! is therefore too large by factors 3! and 2!. Hence the actual number of different arrangements of three S's and two F's is given by: 5! ------ = 10 3! 2! This is also the expression for the binomial coefficient 5_C_2. These coefficients you will find are given on most scientific calculators. So the probability of three successes = 10*(1/6)^3*(5/6)^2 The general expression for binomial probabilities are given by the terms of the binomial expansion of (p+q)^n Here, n = number of trials p = probability of success at each trial (constant) q = probability of failure at each trial (p+q = 1) The probability of r successes is P(r) = nCr*p^r*q^(n-r) In example we did above, n = 5, p = 1/6, q = 5/6, and r = 3: P(3) = 5C3*(1/6)^3*(5/6)^2 = 10*(1/6)^3*(5/6)^2 = 0.03215 -Doctor Anthony, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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