Module 03 Calculating Probabilities

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May 29, 2024

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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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