A and B are independent. Prove the situations.

Question

For events A, B, C $\subset \Omega$ we write A $\bot$ B if A and B are independent.

1) Is it true that if A $\bot$ B and A $\bot$ C then A $\bot$ B $\cap$ C?

2) Is it true that if A $\bot$ B and A $\bot$ C then A $\bot$ B $\cup$ C?

Any hint, suggestion or solution is welcomed.

Thank you

Answer 1

Consider the classical fair dice example. So let $T$ be a random variable with values in $\{1,2,3,4,5,6\}$, each equally likely.

Define $A:=\{1,2,3,4\}$, $B:=\{1,3,5\}$ and $C:=\{2,4,5\}$.

Now $P(A \cap B)=1/3=P(A)P(B)$ and $P(A \cap C)=1/3=P(A)P(C)$, so $A$ is independent of $B$ and $A$ is independent of $C$. But $B \cap C = \{5\}$, such that

$P(A \cap (B \cap C))=P(\emptyset)=0 \neq \frac{1}{6}\frac{4}{6} = P(A)P(B \cap C)$.

Equally, $B \cup C = \{1,2,3,4,5\}$, such that

$P(A \cap (B \cup C))=P(\{1,2,3,4\})=P(A)\neq P(A)P(B \cup C)$.

Thus none of the statement hold in general.

Answer 2

Not in general, no.

For a counterexample, say we have four balls numbered $1,2,3,4$. Number $1$ and $2$ are red (and the other two blue), while number $2$ and $3$ are striped (while the other two have solid color). Now pick one ball at random, let $A$ be "the number on the ball is even", $B$ be "the ball is red" and $C$ is "the ball is striped". As a bonus, even $B$ and $C$ are independent, but given the outcome of any of the three, the remaining two stop being independent.

Answer 3

Consider the following experiment:

We flip a coin 3 times, $T_1, T_2, T_3$. Let $A$ denote the event that flips $T_1 = T_2$. Let $B$ denote the event that $T_2 = T_3$. Finally, let $C$ denote the event that $T_3 = T_1$.

$P(A) = P(B) = P(C) = 1/2$ (the probability of flipping a coin twice and getting the same outcome on both tries).

$P(A \cap B) = P(B \cap C) = P(A \cap C) = 2/8$ (the probability that all three flips give the same result), so these events are pairwise independent.

However, $P(A \cap B \cap C) = P(A \cap B) = 2/8$. Hence $C$ is not independent from $A \cap B$.

What happened here is that $A \cap B \subset C$, and a subset of probability $< 1$ cannot be independent from a subset of itself.

So this gives an example for the first question. (It is the classic example of pairwise independent events that are not mutually independent.)