In
a physics experiment, a photographic plate is placed in front of a beam
of radioactive particles, When a particle hits a plate, it leaves a trace on
the plate. The
number of such traces on a plate at
the end of the experiment has a Poisson distribution. After
a long series of experiments under similar conditions, the probability of a
completely blank plate (no traces) was found to be 0.07.What
should be the most frequently occurring number of traces? (Hint I am not
asking for the mean here)

We have to find first the parameter lambda which defines the distribution.

Poisson dist (lambda, k)=lambda^k*e^(-lambda)/k!

we have Poisson (lambda,0) = 0.07

using 0!=1, we get:

e^(-lambda)=0.07

taking natural logs of both sides:

lambda = -ln(0.07) = 2.659 (I used the LN function in Excel)

The poissn distribution is only defined for natural numbers and 0. the peak is is usually near natural numbers close to lambda and there is only one peak. We can use the poisson function is Excel to calculate for n=0, 1, 2:

to find:

Poisson (2.659, 0)=0.1861

Poisson (2.659, 1)=0.2475

Poisson (2.659, 2)=0.2194

Thus the most frequent occurrence of such events is 1