Using Normal Distribution to Approximate Binomial Probabilities

Using Normal Distribution to Approximate Binomial Probabilities
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  • 0:01 Calculating Odds
  • 0:50 Binomial Probabilities
  • 2:56 Approximating the…
  • 4:21 Relating Binomial &…
  • 5:05 Lesson Summary
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Lesson Transcript
Instructor: Artem Cheprasov
Binomial probabilities describe processes in our world. Learn how to create and interpret a binomial probability distribution graph, and discover how the normal distribution can form a good approximation of the binomial distribution.

Calculating Odds

Many important problems in the real world involve calculating odds. Familiar examples may include estimating your chances at a prize drawing or casino games. Think about betting on a coin flip. Since coins are 2-sided, there are only two possible outcomes for each flip, where one of them can be defined as success and the other one as failure. That is, you can either win or lose the bet.

Well, in this lesson you will learn about binomial probabilities, which involve processes in which all outcomes are characterized by two possible states: success or failure. Namely, we will focus on how binomial probabilities can be approximated by a normal distribution. A normal distribution is a frequency distribution that can be represented by a normal, or bell-shaped, curve.

Binomial Probabilities Defined

The binomial probability distribution, often referred to as the binomial distribution, is a mathematical construct that is used to model the probability of observing r successes in n trials. The formula shown on the screen (see video) is used for this calculation, with p being the probability of success for a single event and q being the probability of failure for a single event.

Coin Flip Formula

In this formula, the expression P(X = r) specifies the probability of a single outcome r out of the set of all the possible outcomes represented by X. In the instance of a coin flip, the possible outcomes are heads and tails, in which case r could be defined as either heads or tails. Also note that q is simply the complement of p, such that q = 1 - p. The (n r) term in parentheses evaluates to the expression shown here.

(n r) term expression

Let's work through an example together since all of this is kind of confusing.

Suppose we want to find the probability of getting 7 questions correct on a 10-question multiple-choice test, where each question has five answer choices. Further assume that the answers are chosen completely at random, with the test taker not even seeing the questions. The solution is as shown on the screen (see video).

In this calculation, the probability of success (p) is 0.2 because there are 5 answer choices for each question, and we assume completely random guessing of the answers. The probability of failure is simply 1 - p = 1 - 0.2 = 0.8. The result comes out to be a very small probability, which you should expect because getting 7 out of 10 questions right by random guessing is highly unlikely.

Quite frankly, you may have even figured this out yourself the one time you completely forgot to study for a quiz and tried to guess your way through it. Who needs math for this? I'm kidding, of course.

Anyways, let's proceed to explaining how binomial probabilities can be approximated by a normal distribution.

Approximating the Binomial Distribution

To that end, let's go back to the process of flipping a coin. Suppose we want to find the probabilities associated with the number of times the coin lands on heads after two tosses. The possible outcomes are shown on the screen (see video).

There is one possible outcome in which the coin can land on heads twice (HH), with a corresponding probability of 1/4; two possible outcomes in which the coin can land on heads once (HT and TH), giving a total probability of 2/4; and one possible outcome of the coin not landing on heads (TT), with a probability of 1/4. A plot of this binomial probability distribution is shown on the screen: (see video).

If we were to flip the coin four times, the corresponding probabilities would be as displayed in the data table here (see video). The plot would look as shown on the screen (see video).

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