MathStat474 - HW 5

Posted on Feb 9. Due: Thu Feb 20.

This homework is all about computing and understanding the concept of conditional probabilities, independence of events, the general product rule, the law of total probability, and the Bayes rule.

Problem 1.

Pollution of the rivers in the United States has been a problem for many years. Consider the following events:

  • A: the river is polluted,
  • B: a sample of water tested detects pollution,
  • C: fishing is permitted.

Assume \(P(A) = 0.3\), \(P(B|A) = 0.75\), \(P(B|A′) = 0.20\), \(P (C|A\cap B) = 0.20\), \(P (C|A′ \cap B) = 0.15\), \(P (C|A\cap B′) = 0.80\), and \(P(C|A′ \cap B′) = 0.90\).

  1. Find \(P(A \cap B \cap C)\).
  2. Find \(P(B′ \cap C)\).
  3. Find \(P(C)\).
  4. 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.

If 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, what is the probability that she will receive a speeding ticket?

If the person 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.

Solve Exercise 4.10.13:

Here is a discrete random variable with its probability mass function.

x 1.25 1.5 1.75 2 2.25
f(x) 0.2 0.4 0.1 0.2 0.1
  1. \(P(X\geq 2)=?\)
  2. \(P(X<1.65)=?\)
  3. \(P(X=1.5)=?\)
  4. \(P(X<1.3 \textit{ or } X>2.1)=?\)

Problem 4.

Suppose that the four inspectors at a film factory are supposed to stamp the expiration date on each package of film at the end of the assembly line. John, who stamps 20% of the packages, fails to stamp the expiration date once in every 200 packages; Tom, who stamps 60% of the packages, fails to stamp the expiration date once in every 100 packages; Jeff, who stamps 15% of the packages, fails to stamp the expiration date once in every 90 packages; and Pat, who stamps 5% of the packages, fails to stamp the expiration date once in every 200 packages.

If a customer complains that her package of film does not show the expiration date, what is the probability that it was inspected by John?

Problem 5.

Solve Exercise 4.10.32:

Determine the mean and variance of the random variable in Exercise 4.10.13, which is here problem number 3.

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