Question

Amongthe2,598,960possible5-cardpokerhands,howmanycontainthefollowing:a) Atleastonecardthatisnotaheart(complementofallhearts).b) Cardsofmorethanonesuit(complementofallthesamesuit).c) Atleastonefacecard(complementofnofacecards).

Answer 1

a) To know the probabilite that atleast one cart is a heart car we have to know that a poker deck have 48 carts and 12 of them are hearts, so the probability that 1 cart is herst will be:

$\begin{gathered} p=\frac{\frac{12}{48}+\frac{12}{47}+\frac{12}{46}+\frac{12}{45}+\frac{12}{44}}{5} \\ p=\frac{0.25+0.255+0.26+0.266+0.27}{5} \\ p=0.26 \end{gathered}$

b) the probability that the carts have more than one suit will be 1 minus the probability that all carts have only one suit so will be:

$\begin{gathered} p=1-(\frac{12}{48}\cdot\frac{12}{47}\cdot\frac{12}{46}\cdot\frac{12}{45}\cdot\frac{12}{44}) \\ p=1-0.001 \\ p=0.999 \end{gathered}$

c) we hace 3 face carts for each suit so in total we have 12 face cart, so the probabiliti that a least one is a face cart is tha same as a heart cart so is

$p=0.26$