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. 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 \(P( B < Y <R)\)?


  1. Consider two boxes, one containing 1 black and 1 white marble, the other 2 black and 1 white marble. A box is selected at random, and a marble is drawn from it at random.

    1. What is the probability that the marble is black?
    2. What is the probability that the first box was the one selected given that the marble is white?


  1. You are on a game show. There are 3 bins in front of you labelled 1, 2, 3, with Bin 1 having 1 white balls and 1 black ball, Bin 2 have 3 White and 1 Black ball and Bin 3 having 5 White and 1 Black ball. Without you seeing, the game show host chooses one of the bins at random (so that Bin 1 gets selected with probability 1/3, the same for Bin 2, 3) and then selects a ball at random from the bin selected.

    1. What is the probability the ball selected by the game show host is White?
    2. Conditional on the event that the ball selected is White, what is the chance the game show host selected bin 3 was selected?


  1. In a small town in early 2020, amongst the voting population: 80% of Republicans, 14% of Democrats and 35% of Independents approve of Trump’s job performance. In this small town: 30% are Democrat, 28% are Republican and 42% are Independent.

    1. What proportion of voters in this small town think the president is doing a good job?
    2. A randomly chosen voter thinks the president is doing a good job. What is the probability this voter is Democrat?