A professor keeps a stock of pens in her drawer, regardless of whether or not they work. When she is seized by an idea, she grabs a pen at random from those in the drawer. If it doesn’t work, she tosses the pen back in the drawer and grabs another one.

The professor has pens of two colours, red and blue. The drawer contains five red pens, of which two work; and three blue pens, of which two work.

Define the sample space Ω = {pens in the draw}, and events R and B for whether the pen is red or blue. Also define events W and W for whether or not the pen works.

a. Draw a sketch of the sample space Ω as a bag of items.

Use colours red and blue to denote red and blue pens, and symbols W and W to denote pens that work and pens that don’t. For example, a red pen that works should be drawn with a red W.

b. For each of the following probabilities, redraw the diagram of Ω and mark on it the event specified. Also say in words what the probability corresponds to, and calculate the probability.

i. P(R)

ii. P(W)
iii. P(R ∩ W)
iv. P(R | W)
v. P(W | R)