Question

1.14 Consider events ArAg, Avon a sample space Ω. (a) Suppose A, c A-... c AN . Evaluate P(AIA)for i < j and for i > (b) Evaluate the set CAnd D1 A (c) Prove/Disprove: N-1 n AN ) = 1.

Answer 1

(a)

For i < j, P(Ai $\cap$ Aj) = P(Ai)

P(Ai | Aj) = P(Ai $\cap$ Aj) / P(Aj) = P(Ai) / P(Aj)

For i > j, P(Ai $\cap$ Aj) = P(Aj)

P(Ai | Aj) = P(Ai $\cap$ Aj) / P(Aj) = P(Aj) / P(Aj) = 1

(b)

$C = \bigcup_{i=1}^{N}A_i = A_1 \cup A_2 \cup.... \cup A_n = A_n$

$D = \bigcap_{i=1}^{N}A_i = A_1 \cap A_2 \cap.... \cap A_n = A_1$

(c)

(i)

$A_1 \subset A_N$ implies $A_1$ and $\bar{ A_N}$ are disjoint sets.

Thus, $A_1 \cup \bar{ A_N} \ne \Omega$

(ii)

$A_1 \subset A_N$ implies $A_1$ and $\bar{ A_N}$ are disjoint sets.

Thus,

$A_1 \cap \bar{ A_N} \ne \phi$

iii)

$P(A_1 \cup \overline{\overline{A_1} \cap A_2}... \cup \overline{\overline{A_{N-1}} \cap A_N})$

$= P(A_1 \cup A_1 \cup \overline{A_2}... \cup A_{N-1} \cup \overline{A_N})$

$= P(A_1 \cup A_3 \cup ...\cup A_{N-1} \cup \overline{A_2}... \cup \overline{A_N})$

$= P(A_{N-1} \cup \overline{A_2} )$ As, $A_2 \subset A_N \Rightarrow \overline{A_N} \subset \overline{A_2}$

$\ne 1$