# How to compute problems involving conditional probabilities

Each evening a man either watches television or reads a book; the probability that he watches television is $$\frac{4}{5}$$, if he watches television, there is a probability of $$\frac{3}{4}$$ that he will fall asleep. If he reads a book, there is a probability of $$\frac{1}{4}$$ that he will fall asleep, (a) Find the probability that he will fall asleep

Here using tree diagram I have found

• $$P(S)=P(T)\cdot P(S/T)+P(B)\cdot P(S/B)$$
• $$P(S)=\frac{4}{5}\cdot \frac{3}{4}+\frac{1}{5}\cdot\frac{1}{4}$$
• $$P(S)=\frac{13}{20}$$

But I have difficulties coming up with aids and methods to use for part (b)

(b) Suppose the man is not very truthful. When asked if he has been asleep there is a probability of only $$\frac{1}{5}$$ that he will admit he has been to sleep and a probability of $$\frac{3}{5}$$ that he will claim to have been asleep when has not been asleep. Find the probability that

• He goes to sleep and admit it
• He goes to sleep but does not admit it
• He does not go to sleep but claims that he has been asleep
• He does not go to sleep and says that he has not been asleep

Compare the answers to (i) and (iii) to find the probability that if he says he has been to sleep one evening then he is, in fact, telling the truth.

(a) Find the probability that he will fall asleep


Here using tree diagramm I have found

$$P(S)=P(T)\cdot P(S/T)+P(B)\cdot P(S/B)$$

$$P(S)=\frac{4}{5}\cdot \frac{3}{4}+\frac{1}{5}\cdot\frac{1}{4}$$

$$P(S)=\frac{13}{20}$$

He goes to sleep and admit it P(S &A)=P(B)×P(S/B)×P(NA/S)+P(TV)×P(S/TV)×P(NA/S)

P(S&A)=1/5×1/4×4/5+4/5×3/4×1/5

P(S &A)=1/100+12/100

P(S &A)=13/100

He goes to sleep but does not admit it

P(S &NA)=P(B)×P(S/B)×P(NA/S)+P(TV)×P(S/TV)×P(NA/S)

P(NS&A)=1/5×1/4×4/5+4/5×3/4×4/5

P(S &A)=4/100+48/100

P(S &A)=13/25

He does not go to sleep but claims that he has been asleep

P(NS &A)=P(B)×P(NS/B)×P(A/NS)+P(TV)×P(NS/TV)×P(A/NS)

P(NS&A)=1/5×3/4×3/5+4/5×1/4×3/5

P(NS &A)=9/100+12/100

P(NS &A)=21/100

He does not go to sleep and says that he has not been asleep

P(NS &NA)=P(B)×P(NS/B)×P(NA/NS)+P(TV)×P(NS/TV)×P(NA/NS)

P(NS&A)=1/5×3/4×2/5+4/5×1/4×2/5

P(S &A)=6/100+8/100

P(S &A)=7/50

Not sure here on the last one but I got 13/34 probability that he told truth