Binomial Distributions

Binomial Distributions – At Least

Learning Objectives

Calculate the probability of at least [latex]x[/latex] successes or, [latex]P(X \ge x)[/latex].

Calculating [latex]P(X \ge x)[/latex] is similar to calculating [latex]P(X \gt  x)[/latex] with only one difference:

  • [latex]P(X \ge x)[/latex], the probability of at least [latex]x[/latex] successes, includes the [latex]x[/latex] value.
  • [latex]P(X \gt  x)[/latex], the probability of more than [latex]x[/latex] successes, does not include the [latex]x[/latex] value.

In other words:

  • [latex]P(X \gt x) = 1- P(X \le x) = 1-\text{BINOM.DIST}(x, n, p, 1)[/latex]
  • [latex]P(X \ge x) = 1- P(X \le x-1) = 1-\text{BINOM.DIST}(x-1, n, p, 1)[/latex]

Why do we use [latex]x-1[/latex] in the above formula?

  • It has to do with what it means to take a complement.
  • When taking a complement, we take all values outside of that sample space.
  • Since [latex]x[/latex] is included in the range, we ‘stop’ at [latex]x-1[/latex] when taking the complement: [latex]P(X \ge x) = 1 - [P(X=0)+P(X=1)+...+P(X=x-1)][/latex]

Using Formulas to Calculate At least (Example)

Let us revisit the hotel example from the previous section to highlight the similarities and differences between ‘more than’ and ‘at least’ calculations.

Example 28.1 – Using the Formula

Problem Setup: A hotel’s records indicate that 65% of its guests are visitors from Canada.

Question: From a random sample of 12 guests, what is the probability that at least 10 of them are from Canada?

Solution: We know the following:

  • [latex]P(\text{at least 10}) = P(x\ge 10) = P(x=10)+P(x=11) + P(x=12)[/latex]
  • [latex]n=12[/latex] and [latex]p=0.65[/latex].

This gives:

  • [latex]P(x=10) = {}_{12}C_{10} \cdot (0.65)^{10} \cdot (1 - 0.65)^{12-10} = \frac{12!}{10!2!} \cdot (0.65)^{10} \cdot (0.35)^2 =66(0.01346)(0.1225) = 0.10885[/latex]
  • [latex]P(x=11) = {}_{12}C_{11} \cdot (0.65)^{11} \cdot (1 - 0.65)^{12-11} = \frac{12!}{11!1!} \cdot (0.65)^{11} \cdot (0.35)^1 =12(0.00875)(0.35) = 0.03675[/latex]
  • [latex]P(x=12) = {}_{12}C_{12} \cdot (0.65)^{12} \cdot (1 - 0.65)^{12-12} = \frac{12!}{12!0!} \cdot (0.65)^{12} \cdot (0.35)^0 =1(0.00569)(1) = 0.00569[/latex]
  • [latex]P(x\ge 10) =  P(x=10)+P(x=11)+P(x=12) = 0.10885+0.03675+ 0.00569 = 0.1513[/latex]

Conclusion: There is an 15.13% chance that at least 10 of them are from Canada.

Using Excel to Calculate At least (VIDEO)

  • Using Excel’s BINOM.DIST() function is much quicker than using the formula shown in the previous section
  • Again, just be careful of which [latex]x[/latex] value to include and don’t forget to take the complement.
  • We will try this out in the next example.

Example 28.2 – Using Excel and a Complement

Problem Setup: Let us now revisit example 28.1 but we will use Excel.

Question: Can you use Excel’s BINOM.DIST to calculate the probability of at least 10 guests being from Canada?

Solution: Click here to download the Excel solution file. Also, see the video below:

Conclusion: Again, here is an 15.13% chance that at least 10 of them are from Canada.

Key Takeaways (EXERCISE)

Key Takeaways: Binomial Distributions – At Least

Your Own Notes (EXERCISE)

  • Are there any notes you want to take from this section? Is there anything you’d like to copy and paste below?
  • These notes are for you only (they will not be stored anywhere)
  • Make sure to download them at the end to use as a reference

License

Icon for the Creative Commons Attribution-NonCommercial 4.0 International License

An Introduction to Business Statistics for Analytics (1st Edition) Copyright © 2024 by Amy Goldlist; Charles Chan; Leslie Major; Michael Johnson is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License, except where otherwise noted.

Share This Book