How do I test whether two events are independent?

Question

I'm trying to understand how to mathematically determine whether two events are independent. This is for introduction to statistics, so the solution and explanation to this is likely very simple, yet there's something I'm not understanding.

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Here's the question from my textbook:

A statistical experiment has 11 equally likely outcomes that are denoted by a, b, c, d, e, f, g, h, i, j, k.

Event A = {b, d, e, j}

Event B = {a, c, f, j}

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If two events, A and B, are independent, then $P(A) = P(A|B)$. Therefore if I want to determine whether the two events are independent, I need to solve for $P(A)$ and $P(A|B)$ to check whether they are equal.

However $P(A|B) = \frac{P(A∩B)}{P(B)}$, and this is where I get stuck, because I can't know which multiplication rule to use to figure out $P(A∩B)$ since I don't yet know whether the events are independent or not?

If the events are independent, then $P(A∩B) = P(A) * P(B)$. Otherwise if they are dependent, then $P(A∩B) = P(A) * P(B|A)$.

I know that I can solve the question from my textbook by looking at the outcomes that intersect between the two events, but is it possible to figure out whether the two events are independent solely by using the formulae above?

Answer 1

In your case $P(A)=4/11$, $P(B)=4/11$ and $A\cap B={j}$, $P(A \cap B)=1/11$. Now you can see whether A and B are independent.

Answer 2

You figure out the independence of the events based on what they produce from the equations. If P(A∩B)=P(A)∗P(B) then the events are independent. If you have P(A) and are missing both P(A∩B) and P(B) then you don't have enough information to answer if the events are independent or not. Basically you need to have enough information to determine independency.

Answer 3

Here's the question from my textbook:

A statistical experiment has 11 equally likely outcomes that are denoted by a, b, c, d, e, f, g, h, i, j, k.

Event A = {b, d, e, j}

Event B = {a, c, f, j}

Therefore $A\cap B = \{j\}$ and ...