Question

Alice is baking a cake. She doesn’t intend to eat it, but tells you half of what you find is yours. She tells the same thing to Bob too. You and Bob each arrive uniformly at random within an hour. (a) What is the probability of you getting more cake than Bob? (b) What is the probability of you missing your chance to get more cake by less than 5 minutes? 2

Answer 1

Answer: a) P = 0.5, b) P = 0.07

Step-by-step explanation:

Hi!

Lets call X₁ the time at which you arrive, and X₂ the time at which Bob arrives. Both are random variables with uniform density in the interval [0, 60] (in minutes). Their joint distribuition is uniform over the square in the image, with value P = 1/(60*60) = 1/3600.

a) For you to get more cake than Bob, you should arrive earlier. This event is A = { X₁ < X₂ }, the shaded triangle in the figure.The area of this event (set) is half the total area of the square, so P(A) = 0.5.

It makes sense, beacuse its equally probable for you or Bob to arrive earlier, as both have uniform density over the time interval.

b) In this case you arrive later than Bob, but less than 5 minutes later. So the event is B = { X₂ < X₁ < (X₂ + 5) } . This is the gray shaded area in b) part of the image. Its area is the difference two triangles (half square - blue triangle), then the probability is:

$P(B) = 0.5 - \frac{(0.5*55^2)}{3600} =0.07$