Question

Compute the probability of being dealt a flush (except straight flush) in poker.

• A. $\frac{{}^{13}{C}_5{}^4{C}_1-40}{{}^{52}{C}_5}$
• B. $\frac{{}^{13}{C}_5{}^4{C}_3-40}{{}^{52}{C}_5}$
• C. $\frac{{}^{13}{C}_1{}^4{C}_1}{{}^{52}{C}_5}$
• D. None of these

Answer 1

The correct option is A $\frac{{}^{13}{C}_5{}^4{C}_1-40}{{}^{52}{C}_5}$

Flush in $5$ cards.

Flush is cards have same symbol

Selecting one symbol from four symbols$={}^4{C}_1$.

$13$ cards will have same symbol.

Select $5$ cards from same symbol $={}^{13}{C}_5$.

Chances of straight flush in one symbol$=5$ cards in a row.

Five numbers occur consecutively ie $10ways$

Total $4$ symbols so $40$ ways of straight flush.

Total number of ways of flush $={}^{13}{C}_5{\times }^4{C}_1-40$

Total number of ways of selecting $5$ cards $={}^{52}{C}_5$
Total probability$=\frac{{}^{13}{C}_5\times {}^4{C}_1-40}{{}^{52}{C}_5}$