Tutorial 3 Questions

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School

University of Toronto *

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Course

236

Subject

Mechanical Engineering

Date

Jan 9, 2024

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pdf

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1

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University of Toronto Department of Mechanical and Industrial Engineering MIE236: Probability (Fall 2023) Tutorial 3 Problem 1: Pollution of the rivers in the United States has been a problem for many years. Consider the following events: ?: the river is polluted, ?: a sample of water tested detects pollution, ?: fishing is permitted. Assume 𝑃(?) = 0.3 , 𝑃(?|?) = 0.75 , 𝑃(?|?′) = 0.20 , 𝑃(?|? ∩ ?) = 0.20 , 𝑃(?|? ∩ ?) = 0.15 , 𝑃(?|? ∩ ?′) = 0.80 , and 𝑃(?|?′ ∩ ?′) = 0.90. a) Find 𝑃(? ∩ ? ∩ ?). b) Find 𝑃(? ∩ ?). c) Find 𝑃(?) . d) Find the probability that the river is polluted, given that fishing is permitted and the sample tested did not detect pollution. Problem 2: Police plan to enforce speed limits by using radar traps at four different locations within the city limits. The radar traps at each of the locations 𝐿 1 , 𝐿 2 , 𝐿 3 , and 𝐿 4 will be operated 40%, 30%, 20%, and 30% of the time. A person who is speeding on her way to work has probabilities of 0.2, 0.1, 0.5, and 0.2, respectively, of passing through these locations. a) What is the probability that she will receive a speeding ticket? b) If she received a speeding ticket on her way to work, what is the probability that she passed through the radar trap located at 𝐿 2 ? Problem 3. Denote by ? , ?, and ? the events that a grand prize is behind doors A, B, and C, respectively. Suppose you randomly picked a door, say A. The game host opened a door, say B, and showed that there was no prize behind it. Now the host offers you the option of either staying at the door that you picked (A) or switching to the remaining door (C). Use probability to explain whether you should switch of not. Problem 4. Find the probability distribution for the number of jazz CDs when 4 CDs are selected at random from a collection of consisting of 5 jazz CDs, 2 classical CDs, and 3 rock CDs. Express your results by means of a formula. Problem 5. From a box containing 4 black balls and 2 green balls, 3 balls are drawn in succession, each ball being replaced in the box before the next draw is made. Find the probability distribution for the number of green balls.
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