8 pokers hands are dealt from a shuffled deck without replacement.
a. Find the probability that at least one of the 8 hands is a heart flush(all five cards are hearts).
Pr(at least one of 8 hands is heart flush) $= 1 -$ Pr(none of eight hands is heart flush) $=$
$ \left( 8\frac{\binom{13}{5}}{\binom{52}{5}}-28\frac{\binom{13}{5}}{\binom{52}{5}}\frac{\binom{8}{5}}{\binom{47}{5}}\right)$
Is this answer correct?
b. Find the expected value and variance of the total number of eight hands which are heart flushes.
Expected value $= 8\dfrac{13 \choose 5}{52 \choose 5}$
Variance $= 8\dfrac{13 \choose 5}{52 \choose 5}\left(1- \dfrac{13 \choose 5}{52 \choose 5}\right)$
Are these answers correct?