Binomial Distribution: Learn It 4

  • Use a binomial distribution to calculate probability
  • Determine if a probability model meets the conditions for a binomial distribution

Binomial Distribution Formula

Let’s revisit the experiment flipping the coin [latex]3[/latex] times and counting the number of tails obtained.

Notice that the number of ways to obtain each number of successes in a binomial experiment increases pretty quickly. If we were to flip [latex]4[/latex] coins, there would be:

  • [latex]1[/latex] way to obtain [latex]0[/latex] tails
  • [latex]4[/latex] ways to obtain [latex]1[/latex] tail
  • [latex]6[/latex] ways to obtain [latex]2[/latex] tails
  • [latex]4[/latex] ways to obtain [latex]3[/latex] tails
  • [latex]1[/latex] way to obtain [latex]4[/latex] tails

There is a formula that lets us compute these probabilities more easily.

binomial distribution formula

For a binomial experiment in which the probability of success is [latex]p[/latex] and there are [latex]n[/latex] trials, the binomial distribution gives the probability of obtaining [latex]x[/latex] successes is

[latex]P(X=x) = \dfrac{n!}{x!(n-x)!} \cdot p^{x} \cdot (1-p)^{n-x}[/latex]

where [latex]\frac{n!}{x!(n-x)!}[/latex] is called “[latex]n \mbox{ choose } x[/latex],” which computes the number of ways to obtain [latex]x[/latex] successes out of [latex]n[/latex] trials.

The exclamation mark is the symbol for a factorial. You won’t need to calculate this because we will be using technology for our computations, but [latex]n![/latex] is the product of all the positive numbers preceding the number [latex]n[/latex].

[latex]n! = n(n-1)(n-2) \cdots (2)(1)[/latex]

For example, [latex]3! =(3)(2)(1) = 6[/latex].

On the experiment flipping the coin [latex]3[/latex] times and counting the number of tails obtained, we found that there were [latex]3[/latex] ways to obtain [latex]1[/latex] tail in [latex]3[/latex] coin flips.To see how this corresponds to the formula, observe that for [latex]n=3[/latex] and [latex]x=1[/latex]:[latex]3\text{ choose }1 = \frac{3!}{1!(3-1)!} = \frac{3!}{1!2!} = \frac{6}{1 \cdot 2} = \frac{6}{2} = 3[/latex]

As mentioned, we will be using technology to compute these probabilities, so you won’t need to worry much about the formula.

Feel free to explore the tool.

You can click on the Find Probabilities tab, input values for [latex]n[/latex], [latex]p[/latex], and [latex]x[/latex], and then select which type of probability you would like to compute from the drop-down menu.

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