Class Exercise 5

Problem 1

Consider the following contingency table that shows incidence of myocardial infraction (MI) for women smokers and women who had never smoked.$$ \begin{array}{l|cc|c} & \text{MI Yes} & \text{MI No} & \text{Totals}\\ \hline \text{Smoker} & 55 & 65 & \\ \text{Never smoked} & 25 & 125 & \\ \hline \text{Totals} & & & \end{array} $$Let $A = \{$woman smoker$\}$ and let $B = \{$woman had an MI episode$\}$.
  1. Determine and interpret with a sentence each of $p(A|B),\; p(B|A)$ and $p(A|B^c)$.
  2. Are $A$ and $B$ independent? Give a quantitative argument.

Problem 2

In the context of drawing a single card from a well-shuffled deck give an example of:
  1. Two disjoint events.
  2. Two independent events.

Problem 3

In the backyard you have a few tall, beautiful irises frequented by hummingbirds, bees and other creatures (your knowledge of insects is too shallow to identify them precisely). Let us name some events when observing the iris patch:
  • $H$ - at least one humming bird present
  • $B$ - at least one bee present
  • $O$ - at least one other creature present
    • In $250$ observations of the iris patch you have observed $H\; 100$ times; $B\; 120$ times; $O\; 75$ times; $H\cap B\; 10$ times; $H\cap O\; 24$ times; $B\cap O\; 44$ times. You have never observed three types of visitors at the iris patch at the same time.
    1. Compute $p(H|B),\; p(H|O)$.
    2. Compute $p(B|H),\; p(B|O)$.
    3. Are the events $H$ and $B$ independent? Justify your answer with a quantitative argument based on the values you computed above.
    4. Are the events $H$ and $O$ independent? Justify your answer with a quantitative argument based on the values you computed above.
    5. Are the events $B$ and $O$ independent? Justify your answer with a quantitative argument based on the values you computed above.

    Problem 4

    spring-peeperSpring peeper is a small chorus frog widespread in southern Quebec. It is know that in a certain area 4% of these frogs are infected with fungus (deadly disease). A field test for this fungus shows positive with probability 0.85 if the frog is infected and shows positive with probability 0.1 even if the frog is not infected.
    1. You test a spring peeper from this area in the field and the test for fungus is positive. What is the probability that the frog is infected with fungus.
    2. Another spring peeper tests negative. What is the probability this frog is nevertheless infected?