Question

When studying radioactive material, a nuclear engineer found that over 365 days,

1,000,000 radioactive atoms decayed to 970,258 radioactive atoms, so 29,742 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.
a. The mean number of radioactive atoms that decay per day is
(Round to three decimal places as needed.)

Answer 1

a) The mean number of radioactive atoms that decay per day is 81.485.

b) 0% probability that on a given day, 50 radioactive atoms decayed.

Explanation:

Poisson distribution:

In a Poisson distribution, the probability that X represents the number of successes of a random variable is given by the following formula:

`P(X = x) = (e^(-\lambda)*\lambda^(x))/((x)!)`

In which

x is the number of sucesses

e = 2.71828 is the Euler number

`\lambda` is the mean in the given interval.

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

29,742 atoms decayed during 365 days, which means that:

`\lambda = (29742)/(365) = 81.485`

The mean number of radioactive atoms that decay per day is 81.485.

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

This is P(X = 50). So

`P(X = 50) = (e^(-81.485)*(81.485)^(50))/((50)!) = 0`

0% probability that on a given day, 50 radioactive atoms decayed.