SOS STAT 230 MIDTERM 1 REVIEW SESSION

By Rishi Gupta

Today’s Topics:Mutual Exclusivity of EventsIndependent Events Baye’s Theorem Conditional ProbabilityDe Morgan’s Laws Counting Arguments

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 Sthe probability function takes in an event from S and gives out its probability

Problem 1.1: (A)List the key properties of a

probability function (B)Explain the difference between

mutually exclusive and independent events

PART A: 1.) 0 ≤ P(Ai) ≤ 1 for all iNote: 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

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)

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

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

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

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

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 whiteJar 2: 900 white, 100 blue Jar 3: 10 green, 990 white

Find:(i) The probability of pulling exactly 2

coloured marbles(ii) Find two different expressions for the

probability that he pulls no white marbles

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

(i) A, B(ii) A, B(iii)A, B

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

Problem 1.5 State the following: (1) The definition of conditional probability (2) The Law of Total Probability (3) Baye’s Theorem

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?

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?

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?

Problem 1.8:Identify the differences between the following: (1) n^k (2) n^n(3) n!(4) n(k)

(5) n choose k

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: (i) Everybody exits at the same village (ii) Nobody gets off at the smallest village (iii)People only get off at even-numbered

villages (iv)Two people exit at one village, and the

other two people exit at a different village from the original two

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’