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# Probabilities, Counting, and Equally Likely Outcomes - Finite Math

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Finite Math
Probabilities, Counting, and Equally Likely Outcomes

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• ### Probabilities, Counting, and Equally Likely Outcomes - Finite Math

1. 1. 2.1Probabilities, Counting, and Equally Likely Outcomes
2. 2. Events An event is a subset of the sample space of an experiment. Ex: The event of an “even number” from the experiment of rolling a die.  Sample space: {1, 2, 3, 4, 5, 6}  “qualifying elements”: {2, 4, 6}
3. 3. Quiz 2.1 #1 Consider the experiment of ﬂipping a coin twice. How many elements are in the event of “ﬂipping at least one head?” (Hint: draw a tree diagram [1.4] and determine the sample space ﬁrst, then determine which elements “qualify”) A. 2 B. 3 C. 4
4. 4. Quiz 2.1 #1 Consider the experiment of ﬂipping a coin twice. How many elements are in the event of “ﬂipping at least one head?” (Hint: draw a tree diagram [1.4] and determine the sample space ﬁrst, then determine which elements “qualify”) A. 2 B. 3 C. 4 Answer: B
5. 5. Two Dice “Box” Method Visual representation of the sample space of two dice roll:
6. 6. Two Dice Box Example Suppose we want to know the number of elements in the event “sum less than or equal to 4.” Here’s how we use this box (Circles represent qualifying elements): First die Second die Therefore, the answer is 6.
7. 7. Quiz 2.1 #2 Consider the experiment of rolling two dice. What is the number of elements in the event “difference between rolls is at least 3?” A. 12 B. 14 C. 16
8. 8. Quiz 2.1 #2 Consider the experiment of rolling two dice. What is the number of elements in the event “difference between rolls is at least 3?” A. 12 B. 14 C. 16 Answer: A
9. 9. Outcomes and Probabilities For any experiment, each outcome is said to have a “probability” or “weight” – the likelihood of that event compared to other ones. The probability of all possible outcomes of an experiment must sum up to 1.
10. 10. Equally Likely Outcomes For some experiments, it is intuitive that all outcomes of the experiment are equally likely. For example, the outcomes {1, 2, 3, 4, 5, 6} from rolling a “fair” die is equally likely. Since the probabilities have to sum up to one, each element has a probability of 1/6.
11. 11. Weighted ProbabilitiesLet’s consider the following experiment: An urn has 2 red, 1 white, and 1 blue balls. Let O1 = red, O2 = white, O3 = blue.  O means Outcome Since the chance of drawing each ball is equally likely, each ball has ¼ chance of being drawn w1 = .5, w2 = .25, w3 = .25  W for weights w1 + w2 + w3 = 1
12. 12. Quiz 2.1 #3
13. 13. Quiz 2.1 #3 Let consider an experiment of drawing a card from a deck of cards. What’s the probability of drawing an Ace? A. 1/12 B. 1/13 C. 1/52
14. 14. Quiz 2.1 #3 Let consider an experiment of drawing a card from a deck of cards. What’s the probability of drawing an Ace? A. 1/12 B. 1/13 C. 1/52
15. 15. Quiz 2.1 #3 Let consider an experiment of drawing a card from a deck of cards. What’s the probability of drawing an Ace? A. 1/12 B. 1/13 C. 1/52 Answer: B
16. 16. Summary Deﬁnition:  event  outcome, weight How to determine the number of elements in an event How to use “Two Dice Box” Equally likely outcomes  Determining probabilities of events with an experiment containing equally likely outcomes.
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