B1. (a) City crime records show that 20% of all crimes are violent and 80% are n

Question

B1. (a) City crime records show that 20% of all crimes are violent and 80% are noni- olent. The city believes that 74% of all crimes are reported. i) The conditional probability of reporting a crime given it is a violent crime is 0.9. Given a crime is committed, find the joint probability that it is both violent and it is reported. i) Hence, complete a table of joint and marginal probabilities as begun be- low. Yes No Total Yes No Total 0.74 0.2 Violent Crime 1.0 i) Is whether a crime is violent or not independent of it being reported or not?

Explanation / Answer

(i) Here we are given that:

P( violent ) = 0.2,
P( reported ) = 0.74
P( reported | violent ) = 0.9

Using bayes theorem, we get:

P( reported and violent ) = P( reported | violent )P( violent ) = 0.9*0.2 = 0.18

Therefore 0.18 is the required probability here.

(ii) The required probability distribution here is obtained as:

(iii) From the above table, we get:

P( violent and reported ) = 0.18

P( violent ) P( reported ) = 0.2*0.74 = 0.148 which is not equal to P( violent and reported )

Therefore a crime is violent is not independent of it being reported or not.

Reported Yes Reported No Total Violent Yes 0.18 0.02 0.2 Violent No 0.56 0.24 0.8 0.74 0.26 1

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