Question

example 2: in 2007, a police department found there were 7,831 incidents of hate crimes that
had a bias motivation based on race, religion, sexual orientation, nationality, or
disability. of these incidents, 1,807 were committed by a \known offender\, where one
or more attributes is known about the offender. 2,428 were committed at the place of
residence of the victim. 1,069 were committed both by a known offender and at the place
of residence of the victim.
a.) if one incident is selected at random, find the probability that the incident was
committed by a known offender.
b.) find the probability that the incident was not committed at the place of residence of
the victim.
c.) find the probability that the incident was either committed by a known offender or at
the place of residence of the victim.
d.) find the probability that the incident was neither committed by a known offender nor
at the place of residence of the victim.

Explanation:

Step1: Define total incidents

Total hate crimes $n(T) = 7831$

Step2: Solve part (a): Probability of known offender

Known offender incidents $n(K) = 1807$. Probability is $\frac{n(K)}{n(T)}$.
$\frac{1807}{7831} \approx 0.2307$

Step3: Solve part (b): Probability not at victim's residence

Residence incidents $n(R) = 2428$. Probability of not residence is $1 - \frac{n(R)}{n(T)}$.
$1 - \frac{2428}{7831} = \frac{7831-2428}{7831} = \frac{5403}{7831} \approx 0.6900$

Step4: Solve part (c): Probability of K or R

Use addition rule: $P(K \cup R) = P(K) + P(R) - P(K \cap R)$.
$P(K)=\frac{1807}{7831}$, $P(R)=\frac{2428}{7831}$, $P(K \cap R)=\frac{1069}{7831}$
$\frac{1807}{7831} + \frac{2428}{7831} - \frac{1069}{7831} = \frac{1807+2428-1069}{7831} = \frac{3166}{7831} \approx 0.4043$

Step5: Solve part (d): Probability of neither K nor R

This is $1 - P(K \cup R)$.
$1 - \frac{3166}{7831} = \frac{7831-3166}{7831} = \frac{4665}{7831} \approx 0.5957$

Answer:

a.) $\frac{1807}{7831} \approx 0.231$
b.) $\frac{5403}{7831} \approx 0.690$
c.) $\frac{3166}{7831} \approx 0.404$
d.) $\frac{4665}{7831} \approx 0.596$