How Do Different Approaches to Conditional Probability Affect Problem Solving?

[mohamed el teir]

Homework Statement

suppose we have 9 balls : 2 red, 3 green, 4 yellow. and we draw 2 balls without replacement, the probability that one of them is red and the other is green is : P(R)P(G\R)+P(G)P(R\G) = (2/9)(3/8)+(3/9)(2/8)

i faced a problem in the textbook which says: the probability that a married man watches a show is 0.4, and the prob. that a married woman watched the same show is 0.5, and the prob. the man watches this show given that his wife does is 0.7. what is the probability that a couple watches the show ?

the answer to this problem is the intersection which is P(W)P(M\W) = 0.5*0.7 = 0.35

my question: why didn't we treat this problem like the first one, i mean giving the answer as P(W)P(M\W)+P(M)P(W\M) ?

Homework Equations

probability of A given B = P(A\B) = P(A∩B) / P(B)

The Attempt at a Solution

my thinking about the solution contradiction is merged with the problem statement to be relevant

 

[andrewkirk]

We can't use the first method in the second problem because we do not know P(W|M): the probability that the woman watches the show given that the man does. Fortunately, we can do it as above, which is even easier.

That might lead one to ask - then why can't we do the first problem in the same way as the second one? The answer is that we can. The calculation using that approach is
Probability of one red and one green = P(G1 | R1) P(R1) where
G1 is the event that exactly one green is picked, and
R1 is the event that exactly one red is picked.

P(R1) is the prob that the first ball is red and the second isn't, plus the prob that the second ball is red and the first isn't. That's equal to:
$$P(R1) =\left(\frac{2}{9}\cdot\frac{7}{8}+\frac{7}{9}\cdot\frac{2}{8}\right)$$
We also have
$$P(G1 | R1)=\frac{3}{7}$$
Multiplying these out, you'll see it gives the same answer as the other method.

However, doing it this way is at least as complex as the way it was done above, so no efficiency is gained by doing it this way.

In probability there are often a number of different ways of approaching a problem. With practice, one develops intuitions about which ways are likely to be the quickest for a given sort of problem.

 

[mohamed el teir]

We can't use the first method in the second problem because we do not know P(W|M): the probability that the woman watches the show given that the man does.

but can't we say that this = P(M∩W) / P(M) and we already have the intersection and P(M) ? by the way the second part of the question was asking about this ( P(W|M) )

 

[andrewkirk]

we already have the intersection

No, we are not given that. The intersection is the answer to the problem, not an input to it.

the second part of the question was asking about this ( P(W|M) )

Once we have done the first part, which is to calculate P(M∩W), we can then calculate P(W|M) in the way you suggest.

 

[mohamed el teir]

No, we are not given that. The intersection is the answer to the problem, not an input to it.

i may have a conflict in something, regarding this :

P(R)P(G\R)+P(G)P(R\G)

is P(R∩G) = P(R)P(G\R) + P(G)P(R\G) as a whole, or is it = P(R)P(G\R) = P(G)P(R\G) as if we are adding P(R∩G) to P(R∩G) giving 2P(R∩G) ?

 

[andrewkirk]

It's the second one
P(R∩G)= P(R)P(G|R) = P(G)P(R|G)
As you guess at the end of your post, the first one gives double the value of the correct answer.

 

[mohamed el teir]

i think I've figured what what was wrong in my thinking about this.

for the colored balls example we could normally treat with their tree diagram as follows:

because red, green, yellow are the all possible outcomes of this experiment, so we can normally say: probability of both green and red is : P(R)P(G|R)+P(G)P(R|G)

while in the married man/woman problem, i was treating with the experiment like this:

and this is wrong because man and woman are not all possible outcomes of this experiment, i mean it is not like a coin which will give you a tail or head, so P(M)P(W|M)+P(W)P(M|W) is wrong because it is based on a wrong tree, while this:

is a right tree and can be used to get the correct values