Biological Physics: Energy, Information, Life

3.1. The probabilistic facts of life[[Student version, December 8, 2002]] 71

Your Turn 3d

Suppose we rolltwodice. What’s the probability that the numbers on the dice add up to 2? To

6? To 12? Think about how you used both the addition and the multiplication rule for this.

Totry out all these concepts, here’s a more complicated example. Suppose you are shooting
arrows into a distant target. Wind currents give random shifts to thexcomponent of your arrows’
arrival locations, and independent random shifts to theycomponent. Suppose that the probability
distributionPx(x)isaGaussian with varianceσ^2 ,and the same is true forPy(y).

Example Find the probability,P(r)dr,that an arrow lands a distance betweenrandr+dr

from the bull’s-eye.

Solution: Toanswer we use both the rules discussed above.ris the length of the

displacement vector:r≡|r|≡

√

x^2 +y^2 +z^2 .First we find the joint distribution,

the probability that thex-component lies betweenxandx+dxandthey-component

lies betweenyandy+dy.The multiplication rule gives this as

Pxy(x, y)dxdy = Px(x)dx×Py(y)dy

=

(

2 πσ^2

)− 2 / 2

e−(x

(^2) +y (^2) )/(2σ (^2) )
×dxdy
≡

(

2 πσ^2

)− 1

e−r

(^2) /(2σ (^2) )
d^2 r. (3.16)
The two Gaussians combined into a single exponential involving only the distance
r.
We’re not done. Many different displacementsrall have the same speedr;tofind
the total probability thatrhas any of these values we must now use the addition
rule. Think about all thervectors with length lying betweenrandr+dr.They
form a thin ring of width dr.The joint probability distributionPxy(r)isthe same for
all theser,since it depends only on the length ofr.Sotosum all the probabilities,
wemultiplyPxybythe total area of the ring, which is its circumference times its
thickness: 2πrdr.Wethusget
P(r)dr=

(

1

2 πσ^2

)

e−r

(^2) /(2σ (^2) )
× 2 πrdr. (3.17)
Figure 3.3 shows this distribution.
Notice two notational conventions used above (see also Appendix A): First, the symbol “≡”
above is a special form of the equal sign, which alerts us to the fact thatr≡|r|is adefinition
(it definesr). We pronounce it “is defined as,” or “equals by definition.” Second, the symbol d^2 r
denotes theareaof a little box in position space; it is not itself a vector. The integral of d^2 rover
aregion of space equals that region’s area.
Your Turn 3e
Find the fraction of all the arrows you shoot that land outside a circle of some radiusR,asa
function ofR 0.