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Math 310 Section 7.1 Probability

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What is a probability? Def. In common usage, the word "probability" is used to mean the chance that a particular event (or set of events) will occur expressed on a linear scale from 0 (impossibility) to 1 (certainty), also expressed as a percentage between 0 and 100% http://mathworld.wolfram.com/Probability.html

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Sample Space Def. The sample space is all the possible results of the scenario being considered.

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Ex. If I flip a penny, and we assume that it cannot land on its edge, then there are two possible results: it lands on heads, or it lands on tails. Thus the sample space consists of the outcomes heads and tails. Notation: S = {H, T}

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Ex. Consider the spinner to the right. What is the sample space (what are the possible results)? A B DC S = {A, B, C, D}

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Outcomes and Events Def. The individual results in a sample space are called outcomes. Events are any set of outcomes (any subset of the sample space).

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Ex. Again consider the spinner to the right. What are the outcomes? What would be an example of an event? A B DC The outcomes are: A, B, C and D. One event would be: E = {A, B} (read “A or B”)

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Experimental probability Def. Experimental probabilities are ratios, expressed as fractions, decimals, or percents, determined by considering results or outcomes of experiments. An experiment is an activity whose results can be observed and recorded.

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Theoretical probability Def. Theoretical probabilities are ratios, expressed as fractions, decimals, or percents, determined by considering results or outcomes under ideal circumstances.

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Exerimental vs. Theoretical Def. Consider the toss of a penny. Under ideal conditions, the penny is not pre-disposed to land on either side, (ie the penny is equally likely to land on either heads or tails). We would say the probability of landing on heads is ½ as is the probability of landing on tails. However, in experiments rarely do you get that exact proportion. (text pg 434)

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Bernoulli’s Theorem Thrm. Law of Large Numbers If an experiment is repeated a large number of times, the experimental probability of a particular outcome approaches a fixed number as the number of repetitions increase. This number is the theoretical probability.

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Actually determining probabilities

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Equally likely events Def. When one outcome is just as likely as another the outcomes are equally likely.

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Probability of equally likely events Thrm. For a sample space S, with equally likely outcomes, the probability of an event A is given by: P(A) = Number of elements of A Number of elements of S Number of elements of S = n(a)/n(S)

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Ex. Consider a coin. The sample space is S = {H, T}. Since the outcome of a head is equally likely as a tail then: P(H) = ½ P(T) = ½

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Ex. Consider the spinner to the right. The sample space is S = {A, B, C, D}. What are the probabilities of the outcomes, A, B, C and D? A B DC P(A) = ¼P(C) = ¼ P(B) = ¼P(D) = ¼

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Ex. Consider the spinner to the right. The sample space is S = {A, B, C, D}. What is the probability of the event A = {B, C}. (note: to use thrm the outcomes must be equally likely.) A B DC P(A) = 2/4 = ½

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Not all outcomes are always equally likely however.

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Ex. Consider the spinner to the right. What is the sample space? What is the probability of the outcome A? S = {A, B} P(A) ≠ ½ A B

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P(A) Property The probability of an event is equal to the sum of the probabilities of the disjoint outcomes making up the event. (note: this is useful for figuring out the probability of an event comprised on outcomes that are not equally likely.)

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Ex. Consider a die. What is the sample space. What are the probabilities of the individual outcomes? What is the probability of rolling an even number? S = {1, 2, 3, 4, 5, 6} P(1) = 1/6, etc. P(E) = P({2, 4, 6}) = P(2) + P(4) + P(6)

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Ex. Consider the spinner to the right. What is the sample space? What is the probability of the event {A, B}? S = {A, B, C} P({A, B}) = P(A) + P(B) = ½ + ¼ = ¾ A BC

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Mutually exclusive events Property If events A and B are mutually exclusive, then P(A or B) = P(A U B) = P(A) + P(B)

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Ex. Consider a die. What is the probability of rolling an even number or a 1? A = {2, 4, 6}, B = {1} P(A or B) = P(A) + P(B) = ½ + 1/6 = 4/6

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Non-Mutually Exclusive Events If A and B are events the P(A or B) = P(A) + P(B) – P(A ∩ B)

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What is P(A ∩ B) P(A ∩ B) is the probability of the intersection of two events.

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Ex. Consider a die. What is the probability of rolling an odd number or a number divisible by 3? A = {1, 3, 5}, B = {3, 6}, A∩B = {3} So P(A or B) = ½ + 1/3 – 1/6 = 4/6 = 2/3

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Complementary Events If A is an event and Ā is its compliment, then P(A) + P(Ā) = 1, P(Ā) = 1 – P(A)

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Ex. Consider a die. What is the probability of not rolling a 6? A = {not 6} = {1, 2, 3, 4, 5} So P(A) = 5/6

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Question Why is it that for any event A, 0 ≤ P(A) ≤ 1.

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Certain and Impossible Events A certain event is one whose probability is 1, while an impossible event is one whose probability is 0. Notation: P(Ø) = 0 (impossible event) P(S) = 1 (certain event)

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Properties of Probability 1. P(Ø) = 0 (impossible event) 2. P(S) = 1 3. For any event A, 0 ≤ P(A) ≤ 1. 4. If A and B are events and A ∩ B = Ø, then P(A or B) = P(A) + P(B). P(A or B) = P(A) + P(B). 5. If A and B are events, then P(A or B) = P(A) + P(B) – P(A∩B) 6. If A is an event, then P(Ā) = 1- P(A)

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