In Chapter 7, we focused on using statistics to calculate probabilities. Using relative frequencies, we were able to calculate the likelihood of events using the history of what has happened in the past. However, in some applications we are interested in counting arrangements of objects in order to calculate the likelihood of a particular arrangement, For instance, suppose we wish to rank applicants for a job from a larger pool. How likely is it that the top five applicants are female? To answer this question, we need to be able to count the number of ways to rank five applicants from a larger pool. This will require us to learn about permutations. Permutations are used to calculate arrangements of objects where each objects are chosen without repetition.
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In Chapter 7, we learned how to calculate probability. This requires us to count outcomes in the sample space. For a business, this might mean counting defective products coming off a production line in a factory.
If you are bottling champagne, it is not realistic to physically check whether every bottle has a good seal. Instead, you might examine a small sample of bottles determine if they have a defective seal. Suppose you examine ten bottles. How likely are you to find one or less defective seals?
This type of problem requires us t learn about counting orderings of objects in a systematic way using permutations and combinations. This chapter is all about counting with permutations and combinations. This information will help you to accomplish this chapter’s objective.
Apply combinations and permutations in applications involving counting.
In earlier sections, we learned how to find probabilities of events and compound events. In this section, we’ll continue building on our knowledge of probability by considering the likelihood of events that are linked to another event that is known to have occurred. For instance, if a person is male, how likely is it that they own a smart phone? Or if a tax return results in a refund, what is the probability that it was examined by the IRS? Questions like this where an event is known to occur and we want to find the likelihood of another event lead us to conditional probabilities.
In section 7.1, we examined the probabilities of individual outcomes in the sample space of and experiment. We could find the probability of each outcome if the outcomes were equally likely by counting the number of outcomes in the sample space. In the case where outcomes were not equally likely, the probabilities could be determined by repeating the experiment many times and recording the relative frequency of each outcome.
In this section we look at collections of outcomes called events. These events contain collections of outcomes from the experiment as well as outcomes created by combining other events. We’ll calculate the likelihood of these events using several probability rules. These basic rules form the foundation for calculating the likelihood of events commonly encountered in business and finance.
How likely is that a product produced by a business is defective? How likely is the closing price of a certain stock to be higher than the closing price on the day before? How likely are you to use less than 2 gigabytes of data per month on your cell phone? Questions like these fall into the realm of probability. They reflect the uncertainty involved in business and finance. If we can answer these questions in some mathematical way, we can use the results to make intelligent decisions.
Probability is simply the likelihood of some event occurring. In this section, we’ll learn how to assign a number from 0 to 1 that reflects how likely an event will occur. A probability of 0 means the event will not occur. A probability of 1 means the event will occur. A probability between 0 and 1 reflects varying degrees to which the event might occur. If the chance of rain is 0.1 (often read as a 10% chance of rain), it probably won’t rain. However, a probability of 0.9 (a 90% chance of rain) indicates that it probably will rain. A 50% chance of rain (probability equal to 0.5) means is just as likely to rain as not rain.
In this section we will define probability and learn how to assign probability to events.
