According to a survey, 63% of murders commited last year were cleared by arrest

Question

According to a survey, 63% of murders commited last year were cleared by arrest or exceptional means. Fifty murders commited last year are randomly selected, and the number cleared by arrest or exceptional means is recorded. When technology is used, use the Tech Help button for further assistance.

(a) Find the probability that exactly 38 of the murders were cleared.

(b) Find the probability that between 35 and 37 of the murderers, inclusive, were cleared.

(c) Would it be unusual if fewer than 20 of the murderers were cleared? Why or why not?

Explanation / Answer

A.

Note that the probability of x successes out of n trials is          
          
P(n, x) = nCx p^x (1 - p)^(n - x)          
          
where          
          
n = number of trials =    50      
p = the probability of a success =    0.63      
x = the number of successes =    38      
          
Thus, the probability is          
          
P =    0.018948682   [ANSWER]

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B.

We get P(35) + P(36) + P(37) in the same way as above.

P(35) = 0.071168482
P(36) = 0.050491153
P(37) = 0.032529728

Thus,

P(35 to 37) = 0.154189363 [ANSWER]

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C.

Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    50      
p = the probability of a success =    0.63      
x = the number of successes =    19      
          
Then the cumulative probability is          
          
P(x <=   19   ) =    0.000289838 < 0.01

Thus, it is RARE, as it has very low probability, even less than 1%.

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