Problem 11BSC

Use the Poisson distribution to find the indicated probabilities.

Radioactive Decay Radioactive atoms are unstable because they have too much energy. When they release their extra energy, they are said to decay. When studying cesium-137, a nuclear engineer found that over 365 days, 1,000,000 radioactive atoms decayed to 977,287 radioactive atoms, so 22,713 atoms decayed during 365 days.

a. Find the mean number of radioactive atoms that decayed in a day.

b. Find the probability that on a given day, 50 radioactive atoms decayed.

Answer :

Step 1 of 1

a)

Given

A nuclear engineer found that over 365 days, 1,000,000 radioactive atoms decayed to 977,287 radioactive atoms, so 22,713 atoms decayed during 365 days.

Let the mean number of radioactive atoms that decayed in a day.

=

=

= 62.227 atoms

Therefore the mean number of radioactive atoms that decayed in a day is 62.227 atoms.

b)

Let the probability that on a given day, 50 radioactive atoms decayed.

The average number of successes within a given region is μ.

Here we know that mean = 62.227 and x = 50.

Then the probability of the poisson distribution is

P(x) =

Then,

P(x = 50) =

P( x = 50) = 0.0155