Difference between the probability that the person selected is a male who said yes vs the probability that a male said yes

Question

I really need help with a basic conditional probability question. I am having trouble with the wording: What is the difference between:

a) Find the probability that the person selected is a male who said yes

b)Find the probability that a male said yes

For part A, would you interpret it as: Given that the person is male, what is the probability that he said yes?

And for part B, would you interpret it as, Given that the person said yes, what is the probability that they are male?

If you guys could help me on this problem, I would greatly appreciate it.

Thanks.

Answer 1

According to you, You can have a male set ($M$) and a female set ($F$). And the solution set can be 'yes' set ($Y$) or no set ($N$).

First question says, given that the solution set selected is $Y$ find the probability that the answer was provided from $M$ set.

Second question is straightforward, you have to find the probability of male set, and given that what is the probability that the answer said is yes.

HTH.

Answer 2

The first question seems most naturally interpreted as "The probability that a person is a male and said yes" ($P(M\cap Y$)) (e.g. Proportion of all respondents who said yes and were also male) and the second as "Probability that at least one male said yes".

Answer 3

A.) You have to pick first male from set of male and female.

Then you have to pick yes from set of yes and no.

So it is P(M|Y).

B.) You have to pick first yes from set of yes and no.

Then you have to pick male from set of male and female.

So it is P(Y|M).

If we consider probability of each of male and female $\frac 12$

And probability of each of yes and no is also $\frac 12$

Then you get same answer in both parts otherwise not.

Answer 4

From a given population,

$\qquad{\small{\bullet}}\;\;$Let $M$ be the event that a selected person is male.

$\qquad{\small{\bullet}}\;\;$Let $Y$ be the event that a selected person said "yes".

Then for part $(a)$,

$\qquad$the probability that the person selected is a male who said "yes"

$\qquad=\text{P}(M\;\text{and}\;Y)$,

and for part $(b)$,

$\qquad$the probability that a male said "yes"

$\qquad =\text{P}(Y|M)$.

Answer 5

I suppose that you can select only male or female persons who said yes or no. Then you have this table:

$\begin{array}{|c|c|c|c|} &\hline M &\overline M & \hline \\ \hline Y& P(Y\cap M)&P(Y\cap \overline M)&P(Y) \\ \hline \overline Y& P(\overline Y\cap M)& P(\overline Y\cap \overline M) & P(\overline Y) \\ \hline &P(M) & P(\overline M)&1\\ \end{array}$

$Y$: A selected person said yes.

$\overline Y$: A selected person said no.

$M$: A selected person is male.

$\overline M$: A selected person is female.

In question a) it is asked for the probability that a selected person is who is male and said yes. This is $P(Y\cap M)=P(M|Y)\cdot P(Y)=P(Y|M)\cdot P(M)$

In question b) it is asked for the probability that a selected male person said yes. Or simlarly: The probability that a person said yes given the person is male: $P(Y|M)=\frac{P(Y\cap M)}{P(M)}$

Answer 6

a) Find the probability that the person selected is a male who said yes

Find the probability that the person is male and the person said yes.   This is requesting you find a joint probability.   $\mathsf P(M\cap Y)$

b)Find the probability that a male said yes

Find the probability that the person said yes when given that the person is male.   This is requesting you find a conditional probability.   $\mathsf P(Y\mid M)$