MATH /stat 304
worksheet II
Axioms of probability
Conditional probability
independence
1.
Carefully state the three axioms of a probability law on a sample space .
2. Let A, B, C be events in a sample space  Using only the axioms prove that
(a) P(A) + P(Ac) = 1.
(b) if 𝐴 ⊂ 𝐵 𝑡ℎ𝑒𝑛 𝑃(𝐴) ≤ 𝑃(𝐵).
(c) 𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴 ∩ 𝐵).
(d) 𝑃(𝐴 ∪ 𝐵) ≤ 𝑃(𝐴) + 𝑃(𝐵).
(e) 𝑃(𝐴 ∪ 𝐵 ∪ 𝐶) = 𝑃(𝐴) + 𝑃(𝐵 ∩ 𝐴𝑐 ) + 𝑃(𝐶 ∩ 𝐴𝑐 ∩ 𝐵𝑐 )
3. Let A and B be events and assume that P(B) ≠ 0. Define conditional probability, P(A|B),
of A given B.
4. Let B be an event in . Prove that 𝑃(∙ |𝐵) is a probability measure on the new sample
space B.
5. Let A and B be events in a sample space Define: “A and B are independent.”
6. Consider the following experiment: A four-sided die is tossed twice and the results X, Y
recorded. Assume that this experiment defines an equiprobable sample space.
Let E be the event “X + Y = 4”.
Let F be the event “minimum of X and Y is 2”.
Calculate each of the following:
(a)
(b)
(c)
(d)
P(E)
P(F)
P(F E)
P( E F)
Are E and F independent events?
2