Question

In: Statistics and Probability

When studying radioactive material, a nuclear engineer found that over 365 days, 1,000,000 radioactive atoms decayed...

When studying radioactive material, a nuclear engineer found that over 365 days, 1,000,000 radioactive atoms decayed to 973,133 radioactive atoms, so 26,867 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.

Thanks in advance!

Solutions

Expert Solution

SOLUTION:

From given data,

When studying radioactive material, a nuclear engineer found that over 365 days, 1,000,000 radioactive atoms decayed to 973,133 radioactive atoms, so 26,867 atoms decayed during 365 days.

Given that,

26,867 atoms decayed during 365 days

that is ,

P = Prob(decay per day)

=( 26867 / 1000000) * (1/365)

P =0.0000736

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

= n p

= 1000000 *0.0000736

= 73.6

74

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

X Normal ()

=  8.58

73.6

  8.58

P[X= 50] = P[ 49.5 < X < 50.5 ]

= P[( 49.5 - ) / < Z <   ( 50.5 - ) / ]

= P[( 49.5 - 73.6 ) / 8.58 < Z <   ( 50.5 - 73.6 ) / 8.58​​​​​​​ ]

= P[( 49.5 - 73.6 ) / 8.58 < Z <   ( 50.5 - 73.6 ) / 8.58​​​​​​​ ]

= P [ -2.81 < Z < -2.69]

= P(Z < -2.69) - P(Z < -2.81)

=0.0036 - 0.0025

= 0.0011

orchestra answered 2 years ago
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