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Chapter 14 Probability Time! What is Probability? A number. Specifically, it represents the long-term likelihood of an event occuring. It is a model for what could happen, not what will happen. It is often expressed as a percentage when talking to people, but when doing calculations it is usually left as a fraction or a decimal. How Do We Find Probability? There are a few different ways to find probabilities. A common one is in situations where there is an equal chance for each event to happen. ◦ The probability of drawing a specific card from a deck is 1/52 (assuming the jokers were removed). ◦ The probability of getting a specific number when you roll a random number cube is 1/6. ◦ The probability of heads on a fair coin is 1/2. How Do We Find Probability? Another way to find probability is to create a fraction of the number of times a specific outcome happens divided by all of the outcomes. ◦ For example, if I have a bag with 20 marbles and 5 of them are red, the probability of drawing a red marble if one is chosen at random is 5/20, or 1/4. ◦ The probability of rolling a prime number on a random number cube is 3/6, or 1/2. How Do We Find Probability? Yet another way to find probabilities is if we are just told them. As in, if we are told that P(A) = .3 in the problem. This method of finding probabilities is probably the easiest. How Do We Find Probability? A similar way to find probability is to take a known probability and use the rules of probability to calculate an unknown probability. For example, if we wanted to know Ac and we know that P(A) = .3, then P(Ac) = 1 – P(A) means that P(Ac) = 1 – .3 = .7. How Do We Find Probability? One more way is that we guess. Mind you, this does not mean guess casually or haphazardly. This means guessing by bringing our life experiences and finely honed intuition to bear. So yeah, this is one more thing in Statistics we use our intuition for. This is called a personal probability. How Do We Find Probability? The final way we will discuss is to take a sample and estimate. ◦ In other words if we want to guess at what our probability of making a free throw in basketball is, we could always just grab a basketball, attempt ten free throws in a row, and then ballpark our true probability based on those ten throws. This is called an experimental probability. Finding Probability Summary Equally likely outcomes Determine the fraction They tell us Using rules to calculate Personal probabilities Experimental probabilities Probability In Action Once we have determined a probability, we need to keep in mind what it stands for. It stands for the likelihood that a particular thing happens. To avoid overusing the word thing, we are going to use some specific language. Basic Terminology Random Phenomena – Random things for which we know all possible outcomes, but we do not know which one will actually happen. Phenomena is the plural of phenomenon. This definition also covers things which have unpredictable outcomes as long as one of the outcomes we consider is some kind of a “miscellaneous” or “other” category. Basic Terminology Trial – This is what we call one instance of the random phenomenon. ◦ ◦ ◦ ◦ One card is drawn from a deck. A random number cube is rolled once. A coin is flipped once. A marble is pulled out of the bag. Multiple trials can be certainly be done, but the term trial refers to just a single instance. ◦ A coin flipped three times in a row is three trials. Basic Terminology Outcome – This is the end result of a trial. ◦ This is the number showing on the random number cube. ◦ This is the side showing on the coin. ◦ This is the particular card drawn from the deck. ◦ This is the color of the marble pulled out of the bag. The exact outcome that will occur is unknown before the trial. Basic Terminology Event – This is an outcome or collection of outcomes that we are interested in. Events are typically labeled with capital letters, starting with A. If you are typing, they will usually be bolded as well. Mr. Sanford rarely bothers with this, and will not expect you to bother with it either. Basic Terminology Complement – The complement of an event is the set of all outcomes that are not part of the event. Since every outcome is either part of an event or not part of an event, when you put an event together with its complement, you get all possible outcomes. This is why we use the word complement. Basic Terminology Disjoint – These are two or more events that cannot happen at the same time. ◦ When you roll a die, you will not get a 3 and a 5 on the same roll. ◦ When you flip a coin you will not get heads and tails. An event and its complement are always disjoint. Another way to say disjoint is mutually exclusive. ◦ This gets said a lot more often than you might realize, so you ought to learn it as well. Basic Terminology Dependent – Two events are dependent if knowing that one happened changes the probability of the other. The probability of getting a ticket for speeding and how much you are speeding by are dependent. ◦ If you are not speeding at all, the probability is very low. ◦ If you are going 15 mph over the speed limit, the probability is higher. Basic Terminology Independent – Two events are independent if knowing one variable turned out does not affect the other. This is a somewhat informal definition. We will define independence a little differently later. Would knowing someone’s favorite color give you any insight into knowing someone’s favorite number? Probability Rules Rule: 0 ≤ P(x) ≤ 1 for each outcome or event. Rule: The sum of all possible outcomes must equal 1. ◦ This does not have to mean that the sum of all events equals 1. It is specifically outcomes. Rule: P(Ac) = 1 – P(A) Probability Rules Rule: When events are disjoint, we can add their probabilities to calculate the probability of the union of those events. ◦ P(A U B) = P(A) + P(B) disjoint) (if A and B are Rule: When events are independent, we can multiply their probabilities to calculate the probability of the intersection of those events. ◦ P(A ∩ B) = P(A)•P(B) independent) (if A and B are Last Thing To determine if a probability assignment is legitimate we need to verify two things. We need to verify that each outcome has a probability between 0 and 1, inclusive. We also need to verify that all of the outcomes add to a total probability of 1. Assignments Chapter 14 homework will be released in three waves. First wave: one problem from 1-5, and problems 10 and 24. Due date is forthcoming. There will be a quiz over chapters 14 and 15 eventually. Bulletpoints will be forthcoming.