# How to calculate the probility of 2 independent events of having same value?

We are learning to calculate the probability of sums and difference of random numbers.

Here is the problem: One athlete knows from past experience that the distances of his javelin throws follow a normal distribution with mean 60 (units) and standard deviation 6 (units). The distances of his hammer throws follow a normal distribution with mean 54 (units) and standard deviation 4 (units). The athlete's performances in the two events are independent.

I know how to calculate probability if his next javelin throw is smaller/greater than hammer throw. I can calculate probability of both javelin/hammer throw is greater or smaller than some distance.

But how to calculate the probability that events (javelin throw and hammer throw) can have same value?

to solve the we are using z test with the statistics table for z values.

*I also changed values if you want you can change again.

• Correct me if I'm wrong, but to me it sounds like this: You let $X$ be the distance of the javelin throw, and you let $Y$ be the distance of the hammer throw. You are then asking for $P(X=Y)$, the probability that the two throws are equal. This probability is $0$, since both variables are continuous. – Mankind May 17 '15 at 23:32
• We are using a continuous distribution model. (This model cannot be right, since the lengths of throws are only measured to a certain accuracy.) However, under the continuous model, the probability that independent random variables are exactly equal is $0$. If we interpret the question as asking for the probability are say within $1$ millimetre of each other, you can find an answer using techniques that you know. – André Nicolas May 17 '15 at 23:34
• Two Comments and my Answer almost simultaneous. Addendum (a day later) to my answer follows up on suggestions by @AndréNicolas and me to consider nonzero probability of 'nearly' equal. – BruceET May 18 '15 at 18:18

The probability that two independent continuous random variables are exactly equal is 0. And, of course, normal random variables are continuous. Continuous random variables put 0 probability on any one point.

A more interesting problem might be the probability that the two variables are within one unit of each other.

For discrete random variables the answer need not be 0. For example, If I roll two fair dice, the probability they will show equal numbers is 1/6.

Addendum (A day later): Here is a brief simulation in R. If you are sufficiently interested in the probability that the two random variables match within a unit or half a unit, you can do the formal analysis and check results with the output of the simulation. (With a million iterations, typically accurate to about 2 significant digits.) What is the distribution of $X - Y?$ The last result is an exact one.

 m = 10^6;  x = rnorm(m, 60, 6);  y = rnorm(m, 54, 4)
mean(x == y)
## 0  # simulation actually discrete, but to enough places to avoid match
mean(abs(x - y) < 1)
## 0.078427  # approx. P{|X-Y| < 1}
mean(abs(x - y) < .5)
## 0.039537

# An EXACT computation:
mu.dif = 60-54;  sd.dif = sqrt(6^2 + 4^2)
diff(pnorm(c(-1,1), mu.dif, sd.dif)) # 'pnorm' is CDF
## 0.07819451