EBK INTRODUCTION TO HEALTH PHYSICS, FIF
EBK INTRODUCTION TO HEALTH PHYSICS, FIF
5th Edition
ISBN: 9780071835268
Author: Johnson
Publisher: VST
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Chapter 9, Problem 9.13P

(a)

To determine

Probability of observing exactly 50 counts in 1min.

(a)

Expert Solution
Check Mark

Answer to Problem 9.13P

Probability of observing exactly 50 counts in 1-min is 0.056.

Explanation of Solution

Given:

True mean counting rate is 50 cpm

Mean counts in 1-min, n¯=50

Exact counts, n=50counts

Formula used:

Standard deviation, σ=n¯

  where,n¯= mean value, and 

Probability in normal distribution, p(n)=1σ2πe (n- n ¯ )22σ2

  where,p(n)= probability of finding exactly nn¯= mean value, and σ= standard deviation

Calculation:

Standard deviation, σ=n¯=50=7.07

Probability for exactly 50 counts

  p(n)=1σ 2πe (n- n ¯ ) 2 2 σ 2 p(50)=17.07 2πe (50-50) 2 2 (7.07) 2 =0.056

Conclusion:

Probabilityof observing exactly 50 count is 0.056.

(b)

To determine

Probability of measuring between 47-57 counts in 1 min.

(b)

Expert Solution
Check Mark

Answer to Problem 9.13P

Probability of measuring between 47-57 cpm is 0.68

Explanation of Solution

Given:

Mean counts, n¯=50 cpm

Measuring lie between, 47<n<50 cpm

Formula used:

Standard deviation, σ=n¯

  where,n¯= mean value, and 

Z-score of normal distribution table, Z=nn¯σ

Calculation:

Standard deviation, σ=n¯=50=7.07

Z-score for n = 43, Z=nn¯σ=43507.07=0.99

Z-score for n = 57, Z=nn¯σ=57507.07=0.99

Using Z-score table of normal distribution

Probability of measuring counts between 47-57 cpm

  P(43<n<57)=P(0.99<Z<0.99)=2[P(Z=0.99)0.5]=2[0.83890.5]=0.68

Conclusion:

Probability of measuring between 47-57 cpm is 0.68.

(c)

To determine

Probability of measuring more than 57 counts in 1min.

(c)

Expert Solution
Check Mark

Answer to Problem 9.13P

Probability of measuring more than 57 counts in 1-min is 0.16.

Explanation of Solution

Given:

True mean counting rate, n¯=50

Exact count rate n liesbetween, 47<n<50

Formula used:

Standard deviation, σ=n¯

  where,n¯= mean value, and 

Z-score of normal distribution table, Z=nn¯σ

Calculation:

Standard deviation, σ=n¯=50=7.07

Z-score for n = 43, Z=nn¯σ=57507.07=0.99

Using the Z-score table of normal distribution

The probability of measuring more than 57 counts in 1min.

  P(n>57)=P(Z>0.99)=[1P(Z=0.99)]=[10.8389]=0.16

Conclusion:

Probability of measuring more than 57 counts in 1-min is 0.16.

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