Homework 3 will be given on Thursday. Some or all of the following questions will be included.


  1. You ask your neighbor to water a sickly plant while you are on vacation. Without water, it will die with probability \(.8\); with water, it will die with probability \(0.15\). You are 90 percent certain that your neighbor will remember to water the plant.

    1. What is the probability that the plant will be alive when you return?
    2. If the plant is dead upon your return, what is the probability that your neighbor forgot to water it?


  1. What is the probability that at least one of a pair of fair dice lands on 6, given that the sum of the dice is \(i\)? Answer this question for \(i= 2,…,12\).


  1. You have two events A and B. You are told \(P(A) = .6,P(B) = .4\) and \(P(A\cap B) = .2\). Find \(P(A \cup B^c)\).


  1. Shankar is teaching 2 courses, let us call them 435 and FYS. 435 has 3 sophomores, 8 juniors and 13 seniors; FYS has 5 sophomores, 7 juniors and 6 seniors. Suppose he wants to understand how his teaching is for the students. He picks one student from each class at random. Find the probability that both students are of the same type (i.e. both sophomores, or both juniors etc). Simplify your answer so it looks like a fraction without any factorials (e.g. 12/25).


  1. Randomly choose a pair \({m,n}\) (with replacement) from the set of consecutive integers \(6,7,8 ...,14\). What is the probability that the difference between \(m, n\) namely \(|m - n|\) is an integer multiple of 4? (Here recall \(|x|\) represents the absolute value of \(x\) so \(|-3| = 3,\ \ \ |3| = 3\). Simplify your answer so it looks like a fraction without any factorials (e.g. \(12/25\)).


  1. You have one five sided die and another eight sided die, both of which are fair (all faces have the same chance of appearing). Your friend will roll both dice and tell you the sum of the faces that show up on the dice. Suppose she says the sum was 6. What is the chance the five sided dice turned up a two?


  1. Find a formula for the event \(P(A|B^c)\) only in terms of \(P(A), P(B)\) and \(P(A \cap B)\).


  1. A red die, a blue die, and a yellow die (all 6-sided) are rolled. We are interested in the probability that the number appearing on the blue die is less than that appearing on the yellow die, which is less than the number appearing on the red die. That is, with \(B\), \(Y\), and \(R\) denoting, respectively, the number appearing on the blue, yellow and red die, we are interest in \(P( B < Y <R)\).

    1. What is the probability that no two of the dice have the same number?
    2. Given that no two of the dice land on the same number, what is the conditional probability that \(B < Y <R\)?
    3. What is \(P( B < Y <R)\)?