Question

recent crime statistics collected over the past year reveal the following figures: 79% of all people arrested were male, 12% of all people arrested were under the age of 18, 4% of all people arrested were female and under the age of 18.
let m represent the event that the person arrested is male, and u18 represent the event that the person arrested is under the age of 18.
part (a) complete the table below. use two decimals in each of your answers.

	m	m	row probabilities
u18
column probabilities

part (b) a person who was arrested in the past year is randomly chosen. what is the probability that this person is male or under the age of 18? enter your answer to two decimals.
part (c) find the probability that the person chosen in (b) is neither male nor under the age of 18. use two decimals in your answer.
part (d) what is the probability of all people arrested in the past year are male and at least 18 years of age? enter your answer as a decimal to two places.
part (e) are the events m and u18 mutually exclusive events? select the most appropriate reason below.
a. m and u18 are mutually exclusive events, because p(m ∩ u18) = p(m)p(u18).
b. m and u18 are not mutually exclusive events, because p(m ∩ u18) ≠ p(m)p(u18).
c. m and u18 are mutually exclusive events, because p(m ∩ u18) = 0.
d. m and u18 are not mutually exclusive events, because p(m ∩ u18) ≠ 0.
e. m and u18 are mutually exclusive events, because p(m ∪ u18) = 0.

Explanation:

Step1: Calculate basic probabilities

Given $P(M)=0.79$, $P(U18) = 0.12$, $P(M^c\cap U18)=0.04$. Then $P(M\cap U18)=P(U18)-P(M^c\cap U18)=0.12 - 0.04=0.08$. Also, $P(M^c)=1 - P(M)=1 - 0.79 = 0.21$. And $P(U18^c)=1 - P(U18)=1 - 0.12 = 0.88$. $P(M\cap U18^c)=P(M)-P(M\cap U18)=0.79 - 0.08 = 0.71$. $P(M^c\cap U18^c)=P(M^c)-P(M^c\cap U18)=0.21 - 0.04 = 0.17$.

Step2: Fill in the table

• For the cell of $U18$ and $M$: $P(M\cap U18)=0.08$.
• For the cell of $U18$ and $M^c$: $P(M^c\cap U18)=0.04$.
• Row - probability for $U18$: $P(U18)=0.12$.
• For the cell of $U18^c$ and $M$: $P(M\cap U18^c)=0.71$.
• For the cell of $U18^c$ and $M^c$: $P(M^c\cap U18^c)=0.17$.
• Row - probability for $U18^c$: $P(U18^c)=0.88$.
• Column - probability for $M$: $P(M)=0.79$.
• Column - probability for $M^c$: $P(M^c)=0.21$.
• The overall probability (bottom - right cell) is $1.00$.

Step3: Calculate $P(M\cup U18)$ for part (b)

Using the formula $P(M\cup U18)=P(M)+P(U18)-P(M\cap U18)$. Substitute $P(M)=0.79$, $P(U18)=0.12$ and $P(M\cap U18)=0.08$. Then $P(M\cup U18)=0.79 + 0.12-0.08=0.83$.

Step4: Calculate the probability for part (c)

The probability that a person is neither male nor under 18 is $P((M\cup U18)^c)$. Since $P((M\cup U18)^c)=1 - P(M\cup U18)$, and $P(M\cup U18)=0.83$, then $P((M\cup U18)^c)=1 - 0.83 = 0.17$.

Step5: Calculate the probability for part (d)

The probability that a person is male and at least 18 years of age is $P(M\cap U18^c)=0.71$.

Step6: Determine mutual - exclusivity for part (e)

Two events $A$ and $B$ are mutually exclusive if $P(A\cap B)=0$. Here, $P(M\cap U18)=0.08
eq0$, so $M$ and $U18$ are not mutually exclusive events.

Answer:

Part (a):

	$M$	$M^c$	Row Probabilities
$U18^c$	$0.71$	$0.17$	$0.88$
Column Probabilities	$0.79$	$0.21$	$1.00$

Part (b): $0.83$
Part (c): $0.17$
Part (d): $0.71$
Part (e): D. $M$ and $U18$ are not mutually exclusive events, because $P(M\cap U18)
eq0$