Hey first I want to start of by saying that this is homework. I'm just looking for guidance. These problems involve using a deck of cards.
I want to find the conditional property that the second card will be an ace given that the first is a heart. I did this by by trying to find P(B/A)
A: heart was the first card B: drawing the ace as the second card P(B/A) = P(A n B) / P(A) I know that the probability of getting a heart as the first card and the ace on the second is 1/52 from a previous problem = (1/52)/(13/52) = (1/13) P(B/A) = (1/13)
Now the second portion of the question asks to find the probability that the second card draws an ace using the law of total probability. I think this is how you implement it
A: Second card is an ace B: First card is not an ace C: First card is an ace P(A) = P(A|C)P(C) + P(A|B)P(B) P(A|C)P(C) = (((4 choose 1)(3 choose 1))/(52 choose 2)) * (4/52) P(A|B)P(B) = (((48 choose 1)(4 choose 1))/(52 choose 2)) * (48/52) P(A) = 386/2783
However the last part of the question asks if they are independent or not, and I know that they are only independent if
A: First card is a heart B: Second card is an ace P(B|A) = P(B)
And after the above calculations they are not independent. However after some further thought I feel like this might be wrong. I think that whether the first card is a heart should be independent of whether you choose an ace as your second draw. If someone could deny or confirm this, it would be great.