Guglielmo Camporese (gool-yell-moe)

AI Researcher at Disney Research · Zurich

guglielmocamporese [at] gmail [dot] com
Guglielmo Camporese
Computing the Probability of Catching a Mosquito by Hands

You're lying in bed trying to fall asleep. A mosquito buzzes near your ear. You swat — it's gone. What were the odds you'd actually catch it?

This is a fun back-of-the-envelope calculation. Let's model it properly.

Body diagram

The model

The relevant variables are:

  • $v_m$ — speed of the mosquito,
  • $v_a$ — speed of your arm,
  • $l_a$ — length of your arm,
  • $d_{es}$ — distance from ear to shoulder.

We place the mosquito exactly at your ear. The moment you decide to swat, it starts flying away in a random direction. After $\nabla t$ seconds your hand reaches the ear position — but the mosquito has already moved to some point on a half-sphere of radius $d_m$.

Mosquito half-sphere diagram
The mosquito escapes into a half-sphere of radius $d_m$ before your hand arrives.

Derivation

The radius of the escape half-sphere is how far the mosquito can travel in the reaction time $\nabla t$:

$$d_m = v_m \cdot \nabla t$$

The reaction time is how long your arm takes to reach your ear from rest, travelling distance $d_{es} + l_a$:

$$\nabla t = \frac{d_{es} + l_a}{v_a}$$

Assuming the mosquito is uniformly distributed over the half-sphere, the probability of catching it inside a ball of radius $h$ (your closed fist) is:

$$ \begin{split} P_{\text{catch}} &= \int_{H\text{-}S(h)} \frac{1}{V(H\text{-}S)} \, dx\,dy\,dz \\ &= \int_0^h \int_0^\pi \int_0^\pi \frac{2}{\frac{4}{3}\pi d_m^3} \, r^2 \sin\phi \, d\theta \, d\phi \, dr \\ &= \left( \frac{h}{d_m} \right)^3 = \left( \frac{v_a \cdot h}{v_m(d_{es} + l_a)} \right)^3 \end{split} $$

Numbers

  • $v_m = 0.56 \pm 0.1 \ \left[\frac{m}{s}\right]$ — average mosquito speed,
  • $v_a = 2.5 \ \left[\frac{m}{s}\right]$ — arm speed of a young adult [1],
  • $l_a = 0.6 \ [m]$ — average arm length,
  • $d_{es} = 0.2 \ [m]$ — ear-to-shoulder distance,
  • $h = 0.04 \ [m]$ — radius of a closed fist.

Plugging in:

$$P_{\text{catch}} = \left( \frac{2.5 \times 0.04}{0.56 \times 0.8} \right)^3 \approx 1.11\%$$

The mosquito will probably fly away :)