The Jacobian for this transformation is
with determinant , hence the area element becomes
Then the integral becomes
where is the unit circle,
so that
Now you could evaluate the integral as-is, but it's really much easier to do if we convert to polar coordinates.
Then
Answer:
The probability that the second card drawn is red, given that the first card was a heart is 0.1225
Step-by-step explanation:
Total cards in a deck = 52
Total red cards = 26
Heart cards = 13
Probability of getting red card =
Total remaining card = 51
Remaining red cards = 25
Probability of getting red card in second draw without replacement =
So, the probability that the second card drawn is red, given that the first card was a heart =
Hence the probability that the second card drawn is red, given that the first card was a heart is 0.1225