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Probability: Liklihood of occurrence; we know the population,

Understand the foundations of probability including terms, laws, and examples. Learn about events, sample space, and probability calculations. Explore key concepts like independence and complements. Improve your grasp of statistical inference and confidence intervals.

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Probability: Liklihood of occurrence; we know the population,

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  1. Probability 2. More terms: Probability: Liklihood of occurrence; we know the population, and we predict the outcome or the sample. Statistics: We observe the sample and use the statistics to describe the unknown population. When we make an inference about the population, it is desirable that we give a measure of ‘confidence’ in our being correct (or incorrect). This is done giving a statement of the ‘probability’ of being correct. Hence, we need to discuss probability.

  2. So, Probability: more terms: event: is an experiment y: the random variable, is the outcome from one event sample space: is the list of all possible outcomes probability, often written P(Y=y), is the chance that Y will be a certain value ‘ y ’ and can be computed: number of ways to succeed p = --------------------------------. Sample Space

  3. 8 things to say about probability: 1. 0  p  1 ; probabilities are between 0 and 1 inclusive. 2. pi = 1 ; the sum of all the probabilities of all the possible outcomes is 1. Event relations 3. Complement: If P(A=a) = p then A complement is is P( A not a) = 1-p (= q sometimes). Note: p + (1-p) = 1 (p+q=1). A complement is also called (not the mean).

  4. 8 things continued 4. Mutually exclusive: two events that cannot happen together. If P(AB)=0, then A and B are M.E. 5. Conditional: Probability of A given that B has already happened. P(A|B) 6. Independent: Event A has no influence on the outcome of event B. If P(A|B) = P(A) or P(B|A) = P(B) then A and B are independent.

  5. 8 things continued again two laws about events: 7. Multiplicative Law: (Intersection ; AND) P(AB) = P(A  B) = P(A and B) = P(A) * P(B|A) = P(B) * P(A|B) if A and B are independent then P(AB) = P(A) * P(B). 8. Additive Law: (Union; OR) P(A B) = P(A or B) = P(A) + P(B) - P(AB).

  6. The Venn diagram below can be used to explain the 8 ‘things’. A AB B

  7. An example: Consider the deck of 52 playing cards: sample space: (A, 2, 3, … , J, Q, K) spades (A, 2, 3, … , J, Q, K) diamonds (A, 2, 3, … , J, Q, K) hearts (A, 2, 3, … , J, Q, K) clubs Now, consider the following events: J= draw a J: P(J)= 4/52=1/13 F = draw a face card (J,Q,K): P(F)= 12/52=3/13 H = draw a heart: P(H)= 13/52

  8. An example.2: Compute the following: 1. P(F complement) = (52/52 - 12/52) = 40/52 2. Are J and F Mutually Exclusive ? No: P(JF) = 4/52 is not 0. 3. Are J and F complement M.E. ? Yes: P(J and ) = 0 4. Are J and H independent ? Yes: P(J) = 13/52 = 1/13 = P(J|H)

  9. An example.3: Compute the following: 5. Are J and F independent ? No: P(J) = 4/52 but P(J|F) = 4/12 6. P(J and H) = P(J) * P(H|J) = 4/52*1/4 = 1/52 7. P(J or H) = P(J) + P(H) - P(JH) = 4/52 + 13/52 - 1/52 = 16/52. End: Probability 2.

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