A card is drawn from a pack, the card is replaced & the pack shuffled. If this is done $6$ times, then find the probability that the cards drawn are $2$ hearts, $2$ diamonds & $2$ black cards?
I cannot think of any other method rather than making cases of sequences in which cards of drawn but that is too lengthy. Could someone please suggest some elegant method or some hint?
Can this question be assumed as "$6$ cards are drawn at random. What is the probability that they are $2$ hearts, $2$ diamonds & $2$ black cards?"
Best Answer
We can use the tried and true method of counting all possible drawings of the 6 cards, then multiplying by the probability that one of these situations happens.
So we first find the number of arrangements of HHDDBB, which is $6!/2^3 = 90$, then we see that the chance of any one of these happening is $(\frac{1}{2})^{10}$. Thus the answer is $\frac{90}{2^{10}}=\frac{45}{2^9}$.