Geometric Probability Examples

You’re sure you can hit a circle on a target with an exploding watermelon being squeezed by rubber bands, so you’ve set up a square target right in the line of fire with a circle in the center. The square is 5 ft long and 5 ft wide and the circle has a radius of 2 ft. What is the geometric probability of some watermelon landing in the circle? Reminder: the area of a circle is πr2.

To find the geometric probability, divide the area of the circle, which is the desired area, by the area of the whole target.

The area of the circle is πr2 or (π)(22), which is (3.14)(4) or 12.56 ft². The area of the whole target is 5 × 5 = 25 ft².

The geometric probability is or 0.5024. Moving the decimal two places to the right and rounding, we get 50.2 %. Now what should we do with the leftover watermelon seeds and target?

A game is programmed to randomly put your character somewhere on the computer screen that has a rectangular lava pit on it. If you land in the lava pit you lose points. The screen is 10 inches by 17 inches and the lava pit is 2 inches by 10 inches. What is the geometric probability that you’ll land in lava?

The geometric probability is the area of the desired region (or in this case, not so desired), divided by the area of the total region.

The area of the lava pit is 2 in × 10 in = 20 in². The area of the screen is 10 in × 17 in = 170 in². That makes the geometric probability or 11.8%. May you have better luck than Darth.

A spit wad machine makes and fires spit wads in a random pattern. What is the probability that a spit wad will hit a 2 ft by 2 ft tile in a ceiling of 50 identical tiles?

The area of the target tile is 2 ft × 2 ft which equals 4 ft². The ceiling has 50 tiles, and since they are identical, each has an area of 4 ft², making the total area of the ceiling 4 ft² × 50 = 200 ft². So the geometric probability is or 2%.

Because we know how many tiles there are, we can also find the probability by dividing the number of tiles in the target, which is 1, by the total number of tiles, which is 50. Luckily math never lies, so also equals 0.02 or 2%. Go, fight, spit.