Question

The data in the following table shows the outcomes of guilty and​ not-guilty pleas in 1039 criminal court cases.

Complete parts​ (a) and​ (b) below.

	Guilty plea	​Not-guilty plea
Sent to prison	399	52
Not sent to prison	574	14

a. What is the probability that a randomly selected defendant either pled not guilty or was sent to​ prison? (round to three decimal places)

b. What is the probability that a randomly selected defendant either pled guilty or was not sent to prison? (round to three decimal places)

Answer 1

Given that the data in the table shows the outcomes of guilty and non-guilty pleas in 1039 criminal court cases. We will make the table to easily solve the question. The table is as follows :

	Guilty plea	Not-guilty plea	Total
Sent to prison	399	52	451
Not sent to prison	574	14	588
Total	973	66	1039

We know, P(A or B) = P(A) + P(B) - P(A and B). ----------(1)

We also know, Probability of any event = (Number of favorable outcomes)/(Total number of possible outcomes) ------(2)

Total number of possible outcomes = 1039. -------(3)

a) Here, we have to find the probability that a randomly selected defendant either pled not guilty or was sent to prison. Thus, let A = criminals who pled not guilty = 66 and B = criminals who were sent to prison = 451. Number of criminals who were sent to prison and pled not guilty = 52.

Thus, P(A) = Probability that the criminals pled not guilty = 66/1039 . [from (2) and (3)]

P(B) = Probability that the criminals were sent to prison = 451/1039 . [from (2) and (3)]

Now, P(A and B) = Probability that the the criminals were sent to prison and pled not guilty = 52/1039 . [from (2) and (3)].

Thus, from (1), P(A or B) = (66/1039)+(451/1039)-(52/1039) = (66+451-52)/1039 = 465/1039 = 0.448(rounded to three decimal places).

Thus, P(pled not guilty or was sent to prison) = 0.448 .

Thus, the probability that a randomly selected defendant either pled not guilty or was sent to prison = 0.448.

b) Now, we have to find the probability that a randomly selected defendant either pled guilty or was not sent to prison. Thus, let A = criminals who pled guilty = 973 and B = criminals who were not sent to prison = 588. Number of criminals who pled guilty and were not sent to prison = 574.

Thus, P(A) = Probability that the criminals pled guilty = 973/1039 . [from (2) and (3)]

P(B) = Probability that the criminals were not sent to prison = 588/1039 . [from (2) and (3)]

P(A and B) = Probability that the criminals pled guilty and were not sent to prison = 574/1039 . [from (2) amd (3)]

Thus, from (1), P(A or B) = (973/1039)+(588/1039)-(574/1039) = (973+588-574)/1039 = 987/1039 = 0.950(rounded upto three decimal places).

Thus, P(pled guilty or were not sent to prison) = 0.950 .

Thus, the probability that a randomly selected defendant either pled guilty or was not sent to prison = 0.950.