Chapter 8 - Binomial and Geometric distributions

Part 1 - Binomials

Binomial distributions have the following defining characterisitics:

(1)Only two mutually-exclusive and complementary events are possible on each trial--success or failure.

(2)The number of trials is fixed (n).

(3)The probability of a success on any trial is fixed at p. This DOES NOT mean that the probability of a success is always 50%.

(4)The trials are independent--knowing one outcome does not help you predict the next.

Always define what X represents, for instance, X = number of daughters (successes).

Shorthand identification for a binomial distribution: Binom(n, p).


The calculator will provide probabilities given n and p: binompdf(n,p[,x]) and binomcdf(n,p[,x]). Use pdf when you want probabilities for individual values of X and cdf when you want cumulative values, like the probability that the number of successes is less than or equal to 5. Insert the X value when you want just one value for a specific value of X. You may omit it when you want all the probabilities. Caution! For binomials, the least value X can take is ZERO, not one, so make sure that you associate the right X values with te correct probabilities.


The formula for P(X=k) = nCk p^k * (1-p)^(n-k).

nCk is "n choose k" or n!/(k!*(n-k)!).

If you calculate these probabilities for each possible value of x from 0 to n and add them up you will get a sum of 1.

The expected value or mean of the number of successes in a binomial setting is "mu sub x" = n*p.

The variance of the number of successes in the binomial setting is sigma squared sub x = n * p * (1-p).

The square root is (of course!) the square root of the variance.


What were those directions for loading binomial values into the lists and graphing as histograms? Use seq(X,X,0,n) --> L1 to populate the Xs and binompdf(n,p) --> L2 to insert the corresponding probabilities. To graph, select the histogram tool, use L1 as the xlist and L2 as the freq. You can use zoom 9 to generate a first stab at the graph. Then fix the graph using the window controls.


Part 2--Geometric Distribution

This was different from the binomial in that we are counting the number of trials UNTIL we achieve success, then we stop. This means that X is the number of trials it took and there is no "n" involved. Theoretically, it could take us infinitely many tries before we had a successful result.

Defining characteristics: fixed p, s/f, independent trials, count until success (not a fixed n).

The expected value of x, the number of trials required, is 1/p, where p is the probability of a success in one try. The variance is (1-p)/p^2.

The probability distribution for x = 1, 2, 3, 4, etc. is p, (1-p)p, (1-p)^2*p, (1-p)^3*p, etc.

What is the probability that it takes more than k attempts before you get a success?