Binomial Distribution

Binomial Distribution

1. Combinations

   • Defintion: The number of ways we can choose r objects from a total of n objects,
   • nCr = n! / (n-r)!r!
   • Example: Find the number of hands of 4 cards which can be dealt from a pack of 10.
     • n = 10 and r = 4
       • 10C4 = 210
         • Therefore there are 210 different possible combinations of hands
   • Properties
     • nC0 = nCn = 1
     • nCr = nC(n-r)

2. Binomial Coefficients

   • Pascals Triangle
   • (p+q)^3 = fourth row on the pascals triangle
     • Row number = power + 1
     • 1p^3q^0 + 3p^2q^1 + 3p^1q^2 + 1p^0q^3
       • With the powers, the p starts at the bracket power and counts down to 0 while the q starts at 0 and counts to the bracket power

3. Binomial Distribution

   • B(n,p)
     • The trail is repeated a fixed number, n, times
       • n trails are independant
         • The probability of success, p
           • The probability of success = p and failure = q
             • p + q = 1
   • A single trail has two outcomes = success and failure
   • If X is the number of sixes rolled is then X~B(10,1/6)
     • P(X=4) = 10C4 * (1/6)^4 * (5/6)^(10-4)
       • 0.543

4. Cumulative Binomial Probability Tables

   • X~B(30,0.35)
     • Find the probability that 7 < X <= 12
       • P(X<=12) - P(X<=7)
         • Look at tables where n = 30 and p = 0.35
           • 0.7802 - 0.1238 = 0.6564

5. Mean and Variance

   • Expected mean = np
   • Expected Variance = np(1-p)

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