Question

Radioactive atoms are unstable because they have too much energy. When they release their extra energy, they are said to decay. When studying a particular radioactive element, it is found that during the course of decay over 365 days, 1,000,000 radioactive atoms are reduced to 979,424 radioactrive atoms.

Find the mean number of radioactive atoms lost through decay in a day.
mean =

Find the probability that on a given day 64 radioactive atoms decayed.
P(X = 64) =

Answer 1

Solution

Back-up Theory

If a random variable X ~ Poisson (λ), i.e., X has Poisson Distribution with mean λ then

probability mass function (pmf) of X is given by P(X = x) = e ^{– λ}.λ^x/(x!) ………….....................................................(1),

where x = 0, 1, 2, ……. , ∞

Values of p(x) for various values of λ and x can be obtained by using Excel Function, Statistical, POISSON …. (1a)

Now to work out the solution,

Part (a)

During the course of decay over 365 days, 1,000,000 radioactive atoms are reduced to 979,424 radioactive atoms. =>

Number of atoms reduced in 365 days

= 1,000,000 - 979,424

= 20576

So, mean number of radioactive atoms lost through decay in a day

= 20576/365

= 56.37 Answer 1

Part (b)

Let X = number of radioactive atoms lost through decay in a day.

Then, X ~ Poisson (λ), where mean = λ = 56.37 [vide Answer 1]

So, probability that on a given day 64 radioactive atoms decayed
= P(X = 64)

= 0.0304 [vide (1a)] Answer 2

DONE