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**Unformatted text preview: **Chapter 4: Probability & Counting Rules Probability : Probability is a likelihood or chance that a certain thing will happen. The thing that is happening is known as the outcome. Events are a set of possible outcomes. When tend to refer to the probability of an event happening. Sometimes we calculate the probability of more than one event or an event that is dependent of another. The set or list of all the possible outcomes is the sample space . P (event(s) ) is used to denote the probability of an event occuring Ex.) We can denote the probabililty of obtaining a head as P(head) Often times a letter is used to denote the event. Ex.) P(T) may represent the probability of obtaining a tail There are different types of probabilities. Classical Probability outcomes total occur can event the ways # # Classical probability uses the fact that all events are equally likely. We rely on what we know to determine the probability. Ex.) Flip a fair coin once, Probability/chance of obtaining a head is . Ex.) Select a card from a fair deck of cards Probability/chance of pulling spade is (13 spades out of 52 cards) Empirical Probability (relative frequency) n f ns observatio of number frequency outcomes actual total occured has event the ways = = # # Events do not have to be equally likely for the empirical approach to probability. The experiments are actually performed instead of relying on what we know. The relative frequency of the experiment is the probability. Ex) Flip a coin 20 times and record the outcomes. Say we had 8 heads and 12 tails Probability/chance of obtaining a head is 20 8 , which reduces to 5 2 There are other types of probabilities. We can calculate probabilities based on previous knowledge. The main ones we are dealing with in this class are classical and empirical approaches. You must know that they exist but you are not required to determine them. It is possible to obtain two different probabilities for the same event. For example, P(head). Under the classical approach, the probability is 2 1 =0.5. But, when we actually flip the coin for the empirical approach the result may vary. We can flip the coin 20 times and record the number of heads as 8. As previously mentioned, the probability is 5 2 = 0.4. The number of trials will affect our probability. Flip the fair coin 1000 times and we may then get 498 heads and 502 tails, which now gives a probability close to 0.5 (actually 0.498). Whenever our sample size increases, the probability obtained from the empirical approach gets closer to the probability obtained from the classical approach. This concept is known as the Law of Large Numbers . Example) Identify the type of probability and calculate the probability In a survey conducted at a local restaurant during breakfast hours, 20 people preferred orange juice, 16 preferred grapefruit juice, and 9 preferred apple juice with breakfast. If a person is selected at random, find the probability that she or he prefers apple juice. selected at random, find the probability that she or he prefers apple juice....

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