Calculating Binomial Probabilities

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The Normal Approximation of the Binomial Distribution
14:58 — by Daniel Schaben

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Key Questions

  • Binomial Formula.
    Suppose a binomial experiment consists of n trials and results in x successes. If the probability of success on an individual trial is P, then the binomial probability is:
    #b(x; n, P) = nCx * p^x * q^(n - x) # ; q = 1- p

    ( A binomial experiment (also known as a Bernoulli trial) is a statistical experiment that has the following properties:

    1. The experiment consists of n repeated trials.
    2. Each trial can result in just two possible outcomes. We call one of these outcomes a success and the other, a failure.
    3. The probability of success, denoted by p, is the same on every trial.
    4. The trials are independent; that is, the outcome on one trial does not affect the outcome on other trials. )

    Example:
    Suppose a die is tossed 5 times. What is the probability of getting exactly 2 fours?
    Solution:
    This is a binomial experiment in which the number of trials is equal to 5, the number of successes is equal to 2, and the probability of success on a single trial is 1/6 or about 0.167. Therefore, the binomial probability is:

    #b(2; 5, 0.167) = 5C2 * (0.167)^2 * (0.833)^3 = 0.161#

  • Do you mean "n choose x"

    if x = 1 then a Bernoulli experiment was conducted.

    if x > 1 then multiple Bernoulli experiments were conducted.

    The calculation of this piece is essential to computing the correct binomial probabilities.

  • Because of Central limit theorem.

    Central limit theorem says that if you have n random variables iid with same mean and std deviation (we call #mu# and #sigma#), and we study Y dafined as
    #Y = Sigma(X_i)#

    As n approaches infinity, Y converge in distribution at a normal distribution with parameter (#mu_Y = n mu, sigma_Y = sigma sqrt(n) #).

    So binomial is the sum of n iid Bernoulli distribution, so CLT is valid and we can say that #Y ~ Bin(n,p) ->^d Norm(np,np(1-p))#

    PAY ATTENTION: in binomial the approx is good when #n>20#, #n p >5# and #n (1-p) >5#

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