Respuesta :
Answer:
(a) The probability that the next burglary in town (if one of the three did it) is the burglary of a residence is 0.4783.
(b) The probability that a burglary of a residence is committed by Becky is 0.3241.
Step-by-step explanation:
Let's denote the events as follows:
A = a burglary was committed by Alex.
B = a burglary was committed by Becky.
C = a burglary was committed by Carl.
R = a burglary is a residence burglary.
Given:
P (A) = 0.55,
P (B) = 0.31,
P (C) = 1 - P (A) - P (B) = 1 - 0.55 - 0.31 = 0.14.
P (R ∩ A) = [tex]0.55\times\frac{1}{3}=0.183[/tex]
P (R ∩ B) = [tex]0.31\times\frac{1}{2}=0.155[/tex]
P (R ∩ C) = [tex]0.14\times1=0.14[/tex]
(a)
The total probability formula is:
[tex]P(X)=P(X\cap Y)+P(X\cap Z)[/tex]
Compute the burglary of a residence as follows:
[tex]P(R)=P(R\cap A)+P(R\cap B)+P(R\cap C)\\=0.1833+0.155+0.14\\=0.4783[/tex]
Thus, the probability that the next burglary in town (if one of the three did it) is the burglary of a residence is 0.4783.
(b)
Compute the probability that a burglary of a residence is committed by Becky as follows:
[tex]P(B|R)=\frac{P(R\cap B)}{P(R)}=\frac{0.155}{0.4783}=0.321[/tex]
Thus, the probability that a burglary of a residence is committed by Becky is 0.3241.
