Answer:
(a) 0.2963
(b)0.6522
(c) 0.50
Step-by-step explanation:
Let,
K = Glasses are in the kitchen,
L = Glasses are in the living room
B = Glasses are in the bedroom
F = Mr. Smith found the glasses
Given:
[tex]P(K) =0.20\\P(L)=0.30\\P(B)=0.50\\P(F|K)=0.80\\P(F|L)=0.60\\P(F|B)=0.40[/tex]
Compute the probability that Mr. Smith found the glasses:
[tex]P(F)=P(F|K)P(K)+P(F|L)P(L)+P(F|B)P(B)\\=(0.80\times0.20)+(0.60\times0.30)+(0.40\times0.50)\\=0.54[/tex]
(a)
Determine the probability that the glasses were in the that they Mr. Smith found the glasses :
Use the conditional probability formula:
[tex]P(A|B)=\frac{P(B|A)P(A)}{P(B)}[/tex]
The probability, P (K|F) as follows:
[tex]P(K|F)=\frac{P(F|K)P(K)}{P(F)}\\=\frac{0.80\times0.20}{0.54}\\=0.2963[/tex]
(b)
Determine the probability that Mr. Smith will not find his reading glasses in the bedroom, given that the glasses are in the kitchen:
[tex]P(B^{c}|F^{c})=\frac{P(F^{c}|B^{c})P(B^{c})}{P(F^{c})} \\=\frac{[1-P(F|B)][1-P(B)]}{1-P(F)} \\=\frac{(1-0.40)(1-0.50)}{(1-0.54)} \\=0.6522[/tex]
(c)
Determine the probability that the reading glasses are in the bedroom, given that Mr. Smith did not find them in the bedroom:
[tex]P(B|(F|B)^{c})=\frac{(1-P(F|B))P(B)}{1-P(F|B)} \\=\frac{(1-0.40)\times0.50}{(1-0.40)} \\=0.50[/tex]
