Question

Prove or disprove:

a.) If 𝑃(𝐴) = 𝑃(𝐵) = 𝑝 then 𝑃(𝐴𝐵) ≤ 𝑝^2 .

b.) If 𝑃(𝐴) = 0 then 𝑃(𝐴𝐵) = 0.

c.) If 𝑃(𝐴̅) = 𝑎 and 𝑃(𝐵̅) = 𝑏 then 𝑃(𝐴𝐵) ≥ 1 − 𝑎 − 𝑏.

Answer 1

Prove or disprove:

a.) If 𝑃(𝐴) = 𝑃(𝐵) = 𝑝 then 𝑃(𝐴𝐵) ≤ 𝑝^2 .

Put B=A, then 𝑃(𝐴) = 𝑃(𝐵) = 𝑝, but 𝑃(𝐴𝐵)=𝑃(𝐴) =𝑝> 𝑝^2 (if p is not equal to 0 or 1). The assertion is disproved.

b.) If 𝑃(𝐴) = 0 then 𝑃(𝐴𝐵) = 0.

This is true, as $AB\subset A$ and 𝑃(𝐴𝐵)≤𝑃(𝐴)=0. Therefore 𝑃(𝐴𝐵)=0.

c.) If 𝑃(𝐴̅) = 𝑎 and 𝑃(𝐵̅) = 𝑏 then 𝑃(𝐴𝐵) ≥ 1 − 𝑎 − 𝑏.

False. If we take a=b=0, we would get 𝑃(𝐴𝐵) ≥ 1 which contradicts to the inequality 𝑃(𝐴𝐵)≤𝑃(𝐴)=0.