Porisma

Worked solution

Link both conditional probabilities through the shared intersection

P(X∣Y)=1/2P(X\mid Y)=1/2, P(Y∣X)=2/3P(Y\mid X)=2/3 and P(X∪Y)=5/6P(X\cup Y)=5/6. Find

(i) P(X)P(X), marks

(ii) P(X∪Y′)P(X\cup Y'). marks

[2017/RI/II/5]

CJC · Review problems · Question 3

Answer

(i) 1/21/2; (ii) 2/32/3.

(i) Express all quantities in terms of P(X)

From the conditional definitions, P(X∩Y)=12P(Y)=23P(X)P(X\cap Y)=\tfrac12P(Y)=\tfrac23P(X). Therefore P(Y)=43P(X)P(Y)=\tfrac43P(X). Substitute into the union: P(X)+43P(X)−23P(X)=5/6P(X)+\tfrac43P(X)-\tfrac23P(X)=5/6, giving P(X)=1/2P(X)=1/2.

How did this part go?

(ii) Recover the intersection and interpret the shading

P(X∩Y)=23(1/2)=1/3P(X\cap Y)=\tfrac23(1/2)=1/3 and P(Y)=43(1/2)=2/3P(Y)=\tfrac43(1/2)=2/3. The source diagram includes everything except the Y-only part. Thus P(X∪Y′)=1−P(Y)+P(X∩Y)=1−2/3+1/3=2/3P(X\cup Y')=1-P(Y)+P(X\cap Y)=1-2/3+1/3=2/3.

How did this part go?

Compare your sketch

Expand diagram

Use the scrollbars or arrow keys to inspect details.

Two overlapping circles X left,Y right inside a rectangle. Blue covers all X and everything outside Y; only Y outside X is white. This reconstructs the source cyan X union Y-complement region; areas are schematic.
Two overlapping circles X left,Y right inside a rectangle. Blue covers all X and everything outside Y; only Y outside X is white. This reconstructs the source cyan X union Y-complement region; areas are schematic.
Two overlapping circles X left,Y right inside a rectangle. Blue covers all X and everything outside Y; only Y outside X is white. This reconstructs the source cyan X union Y-complement region; areas are schematic.

Common mistakes

X union Y′ is not the complement of X union Y; keep the source shaded-region interpretation.

Solution notes

Both source conditional derivations, every union substitution, shaded-region identity, diagram and printed marks retained.

Read the hint