Beginner's Probability Reasoning need advice I am doing Data Science course and it requires some knowledge from probability theory so I am doing some self study to catch up. I have self created this probability exercise and wondering whether reasoning below is correct?
Suppose I want to meet this lady, we all volunteer for a volunteer work happens every month(the work last one month). 


*

*For me, I only attend either 1st half of the month or the second half of the month, the probability of each is 0.5

*For her, she attend only 1 quarter of the month, and it can be any of the 4 quarters and the chance for each is 0.25
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Now I want to know the chance of both of us participating the volunteering work at the same time in any of the month. 
Lets call 


*

*M = the even that I volunteer in a month

*S = the even that she volunteer in a month


Then we have P (S, M) = P (S | M).P(M)
As P(M) = 0.5
and P(S) = 0.25
For P (S | M), I draw the following table:
            |     1st half        |    2nd half
----------------------------------------------------------
            |       0.5           |       0.5
----------------------------------------------------------
1st quarter | 0.5 x 0.25 = 0.125  |   0.5 x 0.25 = 0.125
----------------------------------------------------------
2nd quarter | 0.5 x 0.25 = 0.125  |   0.5 x 0.25 = 0.125
----------------------------------------------------------
3rd quarter | 0.5 x 0.25 = 0.125  |   0.5 x 0.25 = 0.125
----------------------------------------------------------
4th quarter | 0.5 x 0.25 = 0.125  |   0.5 x 0.25 = 0.125
----------------------------------------------------------

So the only time that she and I could have met are:


*

*Me attending 1st half and she attends either the 1st quarter or 2nd quarter

*Me attending the 2nd half and she attends either the 3rd or 4th quarter


Then we have 
P(S|M) = 0.125 + 0.125 + 0.125 + 0.125 = 0.5

So, the chance that I and she both meet at the volunteer work at any month would be:
P(S, M) = 0.5 x 0.5 = 0.25

Is my analysis above correct? Thanks for taking time to review and answer me
 A: Define the independent random variables $S = 1,2,3,4$ and $M=1,2$
The event $\{S=i\}$ represents the lady volunteering in quarter $i$ while $\{M=j\}$ represents you volunteering in half j.
You want to compute the probability of the event $$\{[S=1 \cap M=1] \cup [S=2 \cap M=1] \cup [S=3 \cap M=2] \cup [S=4 \cap M=2]\}$$ which is done as follows:
$$P([S=1 \cap M=1] \cup [S=2 \cap M=1] \cup [S=3 \cap M=2] \cup [S=4 \cap M=2]) = 0.5$$
$$ = P([S=1 \cap M=1]) + P([S=2 \cap M=1]) + P([S=3 \cap M=2]) + P([S=4 \cap M=2]) \ \text{by mutual exclusivity}$$
$$ = P[S=1]P[M=1] + P[S=2]P[M=1] + P[S=3]P[M=2] + P[S=4]P[M=2] \ \text{by independence}$$
$$= 0.125 \times 4 = 0.5$$

To compute the probability she shows up if you go to half $j=1$:
$$P(S = 1,2 | M = 1) \stackrel{Why?}{=} P(S = 1,2) \stackrel{Why?}{=} P(S = 1) + P(S = 2) = 0.25 + 0.25 = 0.5$$

To compute the probability she shows up if you go to half $j=2$:
$$P(S = 3,4 | M = 2) \stackrel{Why?}{=} P(S = 3,4) \stackrel{Why?}{=} P(S = 3) + P(S = 4) = 0.25 + 0.25 = 0.5$$

To compute the probability you are both in half $j=1$:
$$P(S = 1,2, M = 1) \stackrel{Why?}{=} P(M = 1)P(S = 1,2) \stackrel{Why?}{=} 0.5 \times [P(S = 1) + P(S = 2)] = 0.5 \times (0.25 + 0.25) = 0.25$$

To compute the probability you are both in half $j=2$:
$$P(S = 3,4, M = 2) \stackrel{Why?}{=} P(M = 2)P(S = 3,4) \stackrel{Why?}{=} 0.5 \times [P(S = 3) + P(S = 4)] = 0.5 \times (0.25 + 0.25) = 0.25$$
