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High School Mathematics - 2
11.11 Binomial Distributions

Binomial Distributions:
  • Binomial distribution is the discrete probability distribution.
  • Binomial distribution describes the possible number of times that a particular event will occur in a sequence of observations. The event is coded binary that is it may or may not occur.
  • The binomial distribution is used when a researcher is interested in the occurrence of an event, not in its magnitude.
  • All of the trials are independent and have only two possible outcomes, success or failure.
  • The probability of success is the same in every trial.
  • The outcome of one trial does not affect the probability of any future trials.
  • The random variable is the number of success in a given number of trials.
  • The probability of x successes in n independent trials is
    P(x) = C(n,x)pxqn-x
    p is the probability of success of an individual trial
    q is the probability of failure on that same individual trial (p + q = 1)
    The expectation for a binomial distribution is
    E(x) = np
    n is the total number of trials and
    p is the probability of success
If a coin is tossed 4 times, then we may obtain 0, 1, 2, 3, or 4 heads. We may also obtain 4, 3, 2, 1, or 0 tails, but these outcomes are equivalent to 0, 1, 2, 3, or 4 heads. The likelihood of obtaining 0, 1, 2, 3, or 4 heads is, respectively, 1/16, 4/16, 6/16, 4/16, and 1/16.
p = 1/2 Thus, in the example discussed here, one is likely to obtain 2 heads in 4 tosses, since this outcome has the highest probability.
Directions: Answer the following questions. Also write at least 5 examples of Binomial distributions of your own.
Q 1: If a family has 4 children, what is the probability that they have 3 boys and 1 girl?

Question 2: This question is available to subscribers only!

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