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SOS STAT 230 MIDTERM 1 REVIEW SESSION By Rishi Gupta

SOS STAT 230 MIDTERM 1 REVIEW SESSION By Rishi Gupta . Today’s Topics:. Mutual Exclusivity of Events Independent Events Baye’s Theorem Conditional Probability De Morgan’s Laws Counting Arguments .

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SOS STAT 230 MIDTERM 1 REVIEW SESSION By Rishi Gupta

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  1. SOS STAT 230 MIDTERM 1 REVIEW SESSION By Rishi Gupta

  2. Today’s Topics: • Mutual Exclusivity of Events • Independent Events • Baye’s Theorem • Conditional Probability • De Morgan’s Laws • Counting Arguments

  3. The “Sample Space” of an experiment or process is the set of all possible distinct outcomes that can occur on a trial • It is usually denoted S • the probability function takes in an event from S and gives out its probability

  4. Problem 1.1: • List the key properties of a probability function • Explain the difference between mutually exclusive and independent events

  5. PART A: 1.) 0 ≤ P(Ai) ≤ 1 for all i Note: you can end up with a prob. of 0 or 1, but if you have a negative prob. Or a prob. > 1, you’re in trouble! 2.) ∑ P(Ai) = 1

  6. PART B: 1.) Mutually exclusive events CANNOT happen at the same time e.g when rolling a dice, you can’t get 1 and 2 on the same role, hence those events are M.E. 2.) Independent events CAN occur at the same time, but occur independently of one another (don’t affect each other)

  7. Note: two events A, B CANNOT both be independent and mutually exclusive

  8. Example 1.2: Given P(A) = 0.3, P(B) = 0.35, find the probability of the following given A and B are mutually exclusive: (i.) A (ii.) AB (iii.) AUB

  9. Note: A is the complement of A • P(A ) = 1 – P(A) • P(A1UA2U.....A3) = ∑ P(Ai)

  10. Note: If two experiments are definitely not going to affect each other, then events from the 2 experiments will be independent. • i.e. Rolling a dice and flipping a coin at the same; these two things don’t affect each other

  11. Example 1.3: Alex blindfolds himself and reaches into 3 distinct jars 1-at-a-time, pulling a single marble from each jar. The contents of the jar are as follows: Jar 1: 600 red, 400 white Jar 2: 900 white, 100 blue Jar 3: 10 green, 990 white Find: • The probability of pulling exactly 2 coloured marbles • Find two different expressions for the probability that he pulls no white marbles

  12. Theorem: If A and B are Independent, the following events are also independent: • A, B • A, B • A, B

  13. Example 1.4: Given A,B independent, prove that A and B are independent.

  14. Problem 1.5 State the following: • The definition of conditional probability • The Law of Total Probability • Baye’s Theorem

  15. Example 1.6: A family has two dogs, Rex and Rover, and a little boy called Russ. None of them is particularly fond of the postal carrier. Given that they are outside, Rex and Rover have a 30% and 40% chance, respectively, of biting the postal carrier. Russ, if he is outside, has a 15% chance of doing the same thing. Suppose that one and only one of the three is outside when the postal carrier comes. If Rex is outside 50% of the time, Rover 20% of the time, and Russ 30% of the time, what is the probability the postal carrier will be bitten? If the postal carrier is bitten, what are the chances that Russ did it?

  16. Example 1.6: A family has two dogs, Rex and Rover, and a little boy called Russ. None of them is particularly fond of the postal carrier. Given that they are outside, Rex and Rover have a 30% and 40% chance, respectively, of biting the postal carrier. Russ, if he is outside, has a 15% chance of doing the same thing. Suppose that one and only one of the three is outside when the postal carrier comes. If Rex is outside 50% of the time, Rover 20% of the time, and Russ 30% of the time, what is the probability the postal carrier will be bitten? If the postal carrier is bitten, what are the chances that Russ did it?

  17. Example 1.7: A gambler is told that one of three slot machines pays off with probability 1/2 while each of the other two slot machines pays off with probability 1/3 The gambler selects a machine at random and plays twice. What is the probability s/he loses the first time and wins the second? If s/he loses the first time and wins the second what is the probability s/he chose the favourable machine?

  18. Problem 1.8: Identify the differences between the following: • n^k • n^n • n! • n(k) • n choose k

  19. Problem 1.9: There are 4 friends on a train that makes routine stops at six villages. Assume that all 4 friends are equally likely to get off at any village. Find the probabilities of the following: • Everybody exits at the same village • Nobody gets off at the smallest village • People only get off at even-numbered villages • Two people exit at one village, and the other two people exit at a different village from the original two

  20. Problem 2.0: Danny is holding all the letters found in the word “statistics”. He accidently spills them on the sidewalk. If he picks up the letters in a random order and places them on his palm, what is the probability: (a) the letters spell statistics? (b) the word starts and ends with an ‘s’ (c) The word starts and ends with an ‘s’ OR starts with ‘a’ and ends with ‘i’

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