A small problem!

[williamrobertsuk]

Given A,B with P(A)=3/4 and P(B)=1/3, what is the smallest and largest value of P(A∩B). So a as largest and b as smallest.

a≤ P(A∩B) ≤b

I managed to calculate b=0.333
I thought at first that a=0.75 but is wasn't correct.
Any thoughts about what a is?

 

[williamrobertsuk]

Found it!
a=0.083

 

[Dr.Peterson]

Given A,B with P(A)=3/4 and P(B)=1/3, what is the smallest and largest value of P(A∩B). So a as largest and b as smallest.

a≤ P(A∩B) ≤b

I managed to calculate b=0.333
I thought at first that a=0.75 but is wasn't correct.
Any thoughts about what a is?

According to the inequality, a is the smallest possible value (or something less than that), and b is the largest (an upper bound). That differs from what you said in words.

But taking the inequality as the intent, you are right that the least upper bound is 1/3 (attained when B ⊂ A).

Clearly a can't be 0.75, since it must be less than b.

The smallest possible intersection would occur when A ∪ B = U, so that P(A∩B) = P(A) + P(B) - 1 = 3/4 + 1/3 - 1 = 1/12 = 0.083.3... . So you got that right, too.

Did you work this out as I did, using Venn diagrams?