In a standard deck of cards plus 1 additional joker (53 cards), I am trying to calculate the probability of drawing a Three of a Kind and a Joker in the same 5 card hand.
At the moment I know the probability for drawing a three of a kind is:
${^{13}\mathrm C_1}\times{^4\mathrm C_3}\times{^{12}\mathrm C_2}\times({^4\mathrm C_1})^2 ~/~ {^{53}\mathrm C_5}$
I am just wondering how I would add in the joker
(this is for school and any help would very appreciated)
On
The probability for obtaining three from four suits in one from thirteen kinds and two other cards, when selecting five from fifty-three cards is actually:$$\def\C#1#2{\mathop{^{#1}\mathrm C_{#2}}}\dfrac{\C{13}{1}\times\C 43\times\left(\C{12}2\times\C42^2+\C{12}1\times\C41\times\C11\right) }{\C{53}5}$$
Note: those two other cards are either: two from four suits in each of two from twelve kinds, or one from four suits in one from twelve kinds and the joker.
One of the non-3-of-a-kind cards must be the Joker, so in your notation $$\dfrac{\,^{13}C_1 \times \,^{4}C_3\times \,^{12}C_1 \times \,^{4}C_1 \times \,^{1}C_1}{\,^{53}C_5}$$