Geometric probability: probability as a ratio of areas
Probabilities become ratios of areas (or lengths or volumes). Sketch the region first; the algebra usually disappears.
Method · Geometric Probability
Intro
Wikipedia contributor — https://commons.wikimedia.org/wiki/File:Sierpinski_triangle_rule_90.gif · Public Domain · Wikimedia Commons
When the randomness is continuous and uniform over some region — two people picking arrival times in an hour, a point dropped in a square, a stick broken at a random spot — probability is just a ratio of measures (lengths, areas, volumes). The trick is to parametrize the sample space, draw the favorable region, and compute its measure. Most of these problems live or die on whether you sketch the picture; the algebra after that is one ratio.
✓ Intro · expand
Try first (productive failure)
Before the worked example: spend 60 seconds taking your best shot at this.
A guess is fine — being briefly wrong about a problem makes the explanation
land harder when you read it. This appears once per tutorial; skip
if you already know the trick.
60s
✓ Try first · expand
Worked example
Two friends agree to meet at a café between 12:00 and 1:00. Each arrives at a uniformly random time during that hour, independent of the other. Each waits at most 15 minutes for the other to arrive, then leaves. What is the probability they meet?
✓ Worked example · expand
Practice 1 of 3Type a fraction, decimal, or expression — mathjs parses it.
✓ Practice · expand
More examples
A handful of harder problems on the same theme. Click any prompt to reveal the solution sketch.
You snap a wooden ruler in two places, with each break-point drawn independently from a uniform distribution along the length. What is the probability that the resulting 3 pieces can be arranged into a triangle?
$\tfrac{1}{4}$. Without loss of generality let the ruler have length 1 and call the two break-points $x, y \sim \mathrm{Unif}(0, 1)$. The three pieces form a triangle iff no piece exceeds $\tfrac{1}{2}$. Symmetry across the cases $x < y$ vs $x > y$ collapses the favourable region to one quarter of the unit square.
+ More examples · expand
Reflection
When you read a problem with continuous random variables, what’s the cue that tells you to draw a picture and take a ratio of areas rather than set up an integral from scratch? And when the favorable region is awkward (a band, a disk segment), why does it usually pay to compute the <em>complement</em> region instead?
