Choosing cards at random without replacing

Question

Out of 52 cards we choose two without replacing them. Are events {we have at least one heart} and {we do not have any figures} independent?

I have managed to solve that with use of binomial coefficients that is (A first event and B second event respectively):

$$P(A)=1-\frac{ 39 \choose 2}{ 52 \choose 2} \\P(B)=\frac{36 \choose 2}{52 \choose 2} \\ P(A \cap B)=\frac{{{9}\choose{1}}{{27}\choose{1}}}{52 \choose 2}+\frac{{{9}\choose{2}}}{52 \choose 2}$$

but, now without use of calculator how can i check whether they are independent or not? It is difficult, furthermore time consuming to calculate whether $P(A)P(B)=P(A\cap B)$

Answer 1

We have the fact that:

$$P(A)P(B)=P(A\cap B)\iff P(B|A)=P(B)\iff P(A|B)=P(A)$$

if $A, B$ are independent events.

So we can check if $P(B|A)=P(B)$ or $P(A|B)=P(A)$ to see if they are independent.

In this case, this is as simple as considering all card pairs that belong to $B$ say, namely those pairs with no figures $(JQKA)$ and showing that the proportion of number of pairs with at least one heart is the same as without condition $B$ being applied.

Answer 2

Hint. Firstly, $P(A)=1-\frac{39 \choose 2}{52 \choose 2}$, $P(B)=\frac{36 \choose 2}{52 \choose 2}$.

Secondly, use the following expression for easier calculation: $${n \choose 2}=\frac{n(n-1)}{2}$$

Hope it helps.