Intro to Business Statistics: Probability Counting Rules Factorials Permutations Combinations

What This Is

Counting rules, including factorials, permutations, and combinations, are essential in business statistics for making informed decisions. For instance, a marketing manager wants to know the number of possible ways to arrange a product's features in a promotional campaign. Understanding these concepts helps in calculating probabilities, designing experiments, and analyzing data.

Key Formulas & Symbols

• n! = n × (n-1) × (n-2) × ... × 2 × 1, where n is a positive integer, represents the factorial of n.
• P(n, r) = n! / (n-r)!, where n is the total number of items and r is the number of items being chosen, represents the permutation of n items taken r at a time.
• C(n, r) = n! / (r!(n-r)!), where n is the total number of items and r is the number of items being chosen, represents the combination of n items taken r at a time.
• nPr = n! / (n-r)!, where n is the total number of items and r is the number of items being chosen, represents the number of permutations of n items taken r at a time.
• nCr = n! / (r!(n-r)!), where n is the total number of items and r is the number of items being chosen, represents the number of combinations of n items taken r at a time.
• P(n, r) = n × (n-1) × ... × (n-r+1), where n is the total number of items and r is the number of items being chosen, represents the permutation of n items taken r at a time using the formula.
• C(n, r) = n × (n-1) × ... × (n-r+1) / r!, where n is the total number of items and r is the number of items being chosen, represents the combination of n items taken r at a time using the formula.

Step-by-Step Procedure

1. Identify the problem: Determine whether you need to calculate permutations or combinations.
2. Choose the formula: Select the appropriate formula based on the problem (factorial, permutation, or combination).
3. Plug in values: Substitute the given values into the formula.
4. Simplify: Simplify the expression to obtain the final answer.
5. Interpret results: Interpret the results in the context of the problem.

Common Mistakes

• Mistake: Using the permutation formula when the order of selection does not matter.
• Correction: Use the combination formula instead, as it accounts for the fact that order does not matter.
• Mistake: Not considering the total number of items when calculating permutations or combinations.
• Correction: Make sure to include the total number of items in the calculation to avoid undercounting or overcounting.
• Mistake: Misinterpreting the results of a permutation or combination calculation.
• Correction: Ensure that the results are interpreted correctly in the context of the problem, taking into account the total number of items and the number of items being chosen.

Quick Practice Problems

1. A marketing manager wants to know the number of possible ways to arrange 5 product features in a promotional campaign. Calculate the permutation.
   Answer: 5P5 = 5! = 120. The calculation involves finding the factorial of 5, which represents the number of ways to arrange the 5 features.
2. A quality control engineer wants to know the number of possible ways to choose 3 items from a set of 10 items. Calculate the combination.
   Answer: C(10, 3) = 10! / (3!(10-3)!) = 120. The calculation involves using the combination formula to find the number of ways to choose 3 items from a set of 10.
3. A sales manager wants to know the number of possible ways to arrange 4 sales representatives in a meeting. Calculate the permutation.
   Answer: 4P4 = 4! = 24. The calculation involves finding the factorial of 4, which represents the number of ways to arrange the 4 sales representatives.

Last-Minute Cram Sheet

• Factorial: n! = n × (n-1) × (n-2) × ... × 2 × 1
• Permutation: P(n, r) = n! / (n-r)! or P(n, r) = n × (n-1) × ... × (n-r+1)
• Combination: C(n, r) = n! / (r!(n-r)!) or C(n, r) = n × (n-1) × ... × (n-r+1) / r!
• Permutation vs. Combination: Use permutation when order matters, use combination when order does not matter.
• Total number of items: Make sure to include the total number of items in the calculation to avoid undercounting or overcounting.
• Interpret results correctly: Ensure that the results are interpreted correctly in the context of the problem.
• ⚠️ p-value is NOT the probability that H₀ is true – it’s the probability of observing the data (or more extreme) if H₀ is true.
• ⚠️ Use α = 0.05 as the default significance level unless specified otherwise.
• ⚠️ Make sure to check the assumptions of the permutation or combination formula before applying it.
• ⚠️ Use the correct formula for permutations and combinations, as they are different.