Given prob. that married man drinks tea is 0.4, ....

[lea8802]

Hey! I need some help on this problem please!

Probability that a married man drinks tea=.4
Probability that a married woman drinks tea=.6
Probability that the husband drinks tea given his wife does=.5

What is the probability that they both drink tea?
What is the probabilty that the wife drinks tea given her husband does?
What is the probability that neither drink tea?

I don't need the answers I just need to formulas of how to get to the answers! Thank you`

 

[soroban]

Re: Probability

Hello, lea8802!

Probability that a married man drinks tea = 0.4
Probability that a married woman drinks tea = 0.6
Probability that the husband drinks tea given his wife does = 0.5

(a) What is the probability that they both drink tea?
(b) What is the probabilty that the wife drinks tea given her husband does?
(c) What is the probability that neither drink tea?

Let \(\displaystyle H\) = husband drinks tea.
Let \(\displaystyle W\) = wife drinks tea.

We are given: \(\displaystyle \(H)\,=\,0.4,\;P(W)\,=\,0.6,\;P(H|W) \,=\,0.5\)

(a) We want \(\displaystyle P(H\,\cap\,W)\)

Bayes' Theorem: \(\displaystyle P(H|W) \:=\:\frac{P(H\,\cap\,W)}{P(W)}\)

So we have: \(\displaystyle \:0.5\:=\:\frac{P(H\,\cap\,W)}{0.6}\;\;\Rightarrow\;\;P(H\,\cap\,W) \:=\0.5)(0.6)\:=\:0.3\)


(b) We want \(\displaystyle P(W|H)\)

Bayes' Theorem: \(\displaystyle \(W|H) \:=\:\frac{P(W\,\cap\,H)}{P(H)} \:=\:\frac{0.3}{0.4} \:=\:0.75\)


(c) We want \(\displaystyle P(\sim H\,\cap\,\sim W)\)

This is the opposite of \(\displaystyle P(H\,\cup\,W)\)

. . \(\displaystyle P(H\,\cup\,W)\:=\(H)\,+\,P(W)\,-\,P(H\,\cap\,W) \:=\:0.4\,+_\,0.6\,-\,0.3\:=\:0.7\)

Therefore: \(\displaystyle \(\sim H\,\cap\,\sim W)\:=\:1\,-\,0.7\:=\:0.3\)

 

[lea8802]

Thanks you are great!