I have a homework question that reads:
You have a deck of $16$ cards, with:
$4$ cards that are white on both sides;
$7$ cards that are white on one side, black on the other;
$5$ cards that are black on both sides.
The cards are shuffled and randomly flipped. You draw one card from the deck and look only at one side of it.
- (a) Draw a tree diagram with the probabilities of all possible card types. Hint: Your tree should have two levels, one for the color of the top of the card, and one for the color of the bottom card.
- (b) What is the probability that the top of your card is black?
- (c) If the top of your card is black, what is the probability that the bottom is white?
- (d) If the top of your card is white, what is the probability that the bottom is white?
Now, here has been my approach thus far:
first, I figured out how many total white sides and black sides there are:
White sides $= 4 \times 2 + 7 = 15$
Black sides $= 5 \times 2 + 7 = 17$
Total sides $= 32$
Probability for top side being white $= 15/32 = 0.46875$
Probability for top side being black $= 1 - 15/32 = 0.53125$
Now, my problem is there is this $P(A | B)$ formula thing that I'm supposed to use to get the next level ones, but I don't know how to do that, so I skipped the probabilities for the second level in the tree for a bit.
I went on to (c), where I said the probability was $7/12$, or about $0.58333$, because if your top is black, either you have a full black card (of which there are $5$), or you have a mixed card (of which there are $7$). You're looking for the probability of a mixed card, so you have a total of $7$ winning picks out of a total $12$ picks ($7 + 5$). I did a similar thing for (d)
Then I went back and filled in the tree by doing the probability of the top side (say the top is black, which is $0.53125$) times the probability of the bottom for that branch (say bottom is white (top has already been black), so that is $0.58333$). So I went $0.53125 \times 0.58333 = 0.309894$, and did $0.53125 - 0.309894$ to get the probability of a full black card.
My question is, how can I fill out the probability tree by using this $P(A|B)$ thing? I know that the formula for $P(A|B)$ is $P(A \cap B) / P(B)$. The thing is, I was getting an obviously wrong probability.
Let $B$ be the event that the top of the card drawn is black.
Let $A$ be the event that the bottom of the card drawn is white.
$P(A|B) = P(A \cap B) / P(B) = (7/16) / (17/32) = 14/17 = 0.823529\ldots$? what?
I did $7/16$ because that's the percentage of getting a mixed card ($A \cap B$), right? I don't understand
Actually, $7/16$ should be $7/12$ right? Because we don't need to have the $4$ full whites in there. Now, $7/12 \times 17/32 = 0.309894$, which is the correct answer I believe. But isn't it $P(A \cap B) / P(B)$?