Normal Approximation of the Binomial Distribution

Definition

If is a binomial random variable of independent Bernoulli trials, each with a probability of success , and if

then the binomial random variable can be approximated by a normal distribution as follows:

todo this is Central Limit Theorem

Example

Patients suffering from a certain illness have a 50% chance of making a full recovery without medication. If 16 patients come into the hospital one night, and leave without medicine, determine the probability that at least 775 of 1600 patients recover.

solution

  1. Test to see if the normal approximation can be used:

    yes, we can use the normal approximation

  2. Calculate the mean and standard deviation

  1. Calculate the z-score and use a z-score table to determine the probability

(1) 774.5 is the lower boundary of the rectangle centred on 775.

Example

An exam consists of 36 MC questions, each with 5 choices. Idiot did not study for the test, and guesses every question.

  1. What is the probability that Idiot scores exactly 5 questions correct?
  1. What is the probability that Idiot gets 10 or fewer questions correct?


    we can use a normal approximation.

(1) 10.5 is the upper bound of the rectangle centred on 10

Example

A six sided die is rolled, determine the probability of rolling:

  1. Exactly 3 sixes on 28 rolls

  1. Exactly 100 sixes on 598 rolls
    Note that is too big a number for most calculators.




    we can use a normal approximation.