Module 03 Calculating Probabilities

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Module 03 Calculating Probabilities Module 03 Calculating Probabilities  Calculating Probabilities Counting Methods for Calculating Probabilities The sample space of events is important to calculate probability. When events have a smaller number of outcomes, it’s fairly easy to write out all of the possibilities, such as the number of outcomes of rolling a six-sided die. However, there are some events where the number of possible outcomes is far too large to calculate by writing them all out.

For example, finding the number of ways to arrange 12 books on a shelf. If you were to write out all of the possible arrangements, you would need to include 479,001,600 different arrangements of 12 different books. This involves far too many outcomes to write by hand. Hence, statistics have created methods to calculate how an event can occur using different counting methods. Factorials Before studying permutations and combinations, it’s crucial to understand how to use a new operation: factorials. A factorial is symbolized by ! and denotes the product of decreasing positive whole numbers. For example, Factorials will be used in the formulas for calculating both permutations and combinations (Triola & Iossi, 2018).

Permutations Permutations is a counting method used when trying to determine how many arrangements are possible when order matters and items cannot be repeated. For example, arranging the first three letters of the alphabet would result in 6 different permutations: abc, acb, bac, bca, cab, cba. When there are different items which are available, and of them are selected without replacement, the number of different permutations is given by (Triola & Iossi, 2018). Examples of Permutations Eight people are running a race. Examples 1 - If 1 st , 2 nd , and 3 rd place are only given awards, how many ways can the awards be given? Permutations are used because order matters (different placement), and each runner can only be awarded one position (no repetition). There are 8 total runners, and we are picking the top 3 places. This will make and n = 8 and r = 3. If we plug this into the permutation formula above, we find the following:

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This means there are 336 different ways that the first 3 runners can be awarded. Examples 2 - How many ways can all eight places be awarded? This is also permutations because order matters (different placement), and each runner can only be awarded one position (no repetition). However, we are placing all 8 runners into all 8 positions. This will make and n = 8 and r = 3. If we plug this into the permutation formula above, we find the following: Note: 0! Is a special calculation, which is equal to one. This means there are 40,320 different ways that 8 players can be placed into 8 positions.

Combinations Combinations is a counting method used when trying to determine how many arrangements are possible when order does not matter , and items cannot be repeated. For example, arranging the first three letters of the alphabet would result in 6 of the same combinations: abc, acb, bac, bca, cab, cba. These would all be considered 1 combination since order does not matter. When there are n different items which are available and r of them are selected without replacement, the number of different combinations is given by (Triola & Iossi, 2018).

Example - Combinations At a local pizzeria, there are 10 possible toppings. How many ways are there to choose 3 different toppings? Combinations are used because the order does not matter (the order in which the toppings are placed are irrelevant) and each topping has to be different (no repetition). There are 10 total toppings, and we are choosing 3 of them. This will make and n = 10 and r = 3. If we plug this into the combination formula above, we find the following: This means there are 120 different 3-topping pizzas that can be made from 10 possible toppings. Probability - Permutations and Combinations Once you understand how to calculate the number of ways an event can occur, you can use these outcomes to calculate the probability of an event. Example - Permutations and Combinations In a certain state’s lottery, there are 48 balls placed in a machine labeled 1 through 48. A player wins if the six numbers drawn match their ticket (in any order). What is the probability that a random person wins the lottery if they purchased a single lottery ticket? In this problem, we need to begin by figuring out the number of ways a player can win. There is only 1 winning combination of six numbers, so there’s only 1 outcome that will win. The second part would be figuring out the total number of possible outcomes possible for picking six numbers from 48 balls. Since order does not matter and you cannot repeat numbers, this would be

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Putting these values together would give us the probability of winning as shown below. Essentially, this shows that winning the lottery would be very, very unlikely since it is so close to zero (impossible). Additional Reading – Permutations and Combinations These counting methods have been around for centuries. To learn more on the history of permutations and combinations and how to calculate their probabilities, read the following on Permutations & Combinations. References Latterell, C.M. (2013). Permutations & combinations. Encyclopedia of Math & Society https://search.ebscohost.com/login.aspx? direct=true&AuthType=ip,shib&db=sch&AN=109435024&site=eds- live&custid=s9076023 Triola, M. F., & Iossi, L. (2018). Elementary statistics (13 th ed.). Pearson.

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