So for whatever reason I just can't wrap my head around this, I know I am doing it wrong.
Question: Two cards are chosen from a pack of cards without replacement. Are the following events independent? (i)the first card is a heart, (ii)the second card is a picture card.
For these events to be independent $P(i \cap ii) = P(i)P(ii)$ and $P(i|ii) = P(i)$ and $P(ii|i) = P(ii)$.
In this case $P(i)={13\over 52}$ and $P(ii)= {12\over 51}$
So $P(i|ii) = {P(i\cap ii)\over P(ii)} = {{3\over 52} \over {12\over 51}} = {51\over 208} \not= {13\over 52}$
According to the book, these events are independent, but I get that they are not. I need help understanding this. What's wrong with my approach?
Best Answer
$P(ii)=\frac {12}{52}$ because it is not conditioned on what the first card was. $P(ii|i)$ is still $\frac 3{13}$-If you go through the chance that the first card is or is not a picture card you will find that. Yes, they are independent because the density of picture cards among the hearts is the same as the density of picture cards among the rest of the deck.