Each thing he named will have a value
H stands for hypothesis, E stands for evidence
1. Club is unwitting participant in mharmans plot (H1) (33%)
2. Club is witting participant (H2) (33%)
3. There is no conspiracy all is coincidence. (H3) (33%)
(Each has 33% because the possibility is split in thirds between all three)
If club flips guilty (E1) (50%)
If club he flips innocent (E2) (50%)
If mharman flips guilty (E3) (50%)
If mharman flips innocent (E4) (50%)
I'm going to go through each hypothesis given each evidence
P(H1|E1)=P(E1|H1)/P(E1)*P(H1)
plug in
P(E1|H1) = 0 Because club can't be both unwitting and guilty
P(H1|E1)=0/.5*.33= 0%
-------------------------
P(H1|E2)=P(E2|H1)/P(E2)*P(H1)
plug in
P(E2|H1) = 50% because he's either unwitting or coincidence
P(H1|E2)=.5/.5*.33 = 33%
-----------------------
P(H1|E3)=P(E3|H1)/P(E3)*P(H1)
plug in
P(E3|H1) = 50% (you get the gist by now)
P(H1|E3)=.5/.5*.33 = 33%
-------------------------
P(H1|E4)=P(E4|H1)/P(E4)*P(H1)
plug in
P(E4|H1) = 0%
P(H1|E4)=0/.5*.33 = 0%
----------------------------
P(H2|E1)=P(E1|H2)/P(E1)*P(H2)
plug in
P(E1|H2) = 50%
P(H2|E1)=.5/.5*.33 = 33%
----------------------------
P(H2|E2)=P(E2|H2)/P(E2)*P(H2)
plug in
P(E2|H2) = 0%
P(H2|E2)=0/.5*.33 = 0%
----------------------------
P(H2|E3)=P(E3|H2)/P(E3)*P(H2)
plug in
P(E3|H2) = 100%
P(H2|E3)=1/.5*.33 = 66%
----------------------------
P(H2|E4)=P(E4|H2)/P(E4)*P(H2)
plug in
P(E4|H2) = 0%
P(H2|E4)=0/.5*.33 = 0%
----------------------------
P(H3|E1)=P(E1|H3)/P(E1)*P(H3)
plug in
P(E1|H3) = 50%
P(H3|E1)=.5/.5*.33 = 33%
----------------------------
P(H3|E2)=P(E2|H3)/P(E2)*P(H3)
plug in
P(E2|H3) = 50%
P(H3|E2)=.5/.5*.33 = 33%
----------------------------
P(H3|E3)=P(E3|H3)/P(E3)*P(H3)
plug in
P(E3|H3) = 0%
P(H3|E3)=0/.5*.33 = 0%
----------------------------
P(H3|E4)=P(E4|H3)/P(E4)*P(H3)
plug in
P(E4|H3) = 100%
P(H3|E4)=1/.5*.33 = 66%
TL;DR
So the stuff under evidence 1 or 2 is what we learn from a club lynch, evidence 3 and 4 stuff is what we learn from a mharm lynch
Club: 0% unwitting OR 33% unwitting, 0% witting OR 33% witting, 33% coincidence
Mharman: 0% unwitting OR 33% unwitting, 0% witting OR 66% witting, 0% coincidence OR 66% coincidence
And basically, the closer the number is to 50%, the less useful it is, the closer it is to 0% or 100%, the more useful it is
So, like Wylted said, we learn a bit more from a Mharman lynch
