Chapter 10, Problem 4RE

Solution

To determine: The probability that a 5-card poker hand will contain all hearts.

The probability that a 5-card poker hand will contain all hearts.is 0.05%.

Given information:

A standard deck of cards contains 52 cards, out of which 26 are red and 26 are black, 13 are of each suit (hearts, diamonds, spades, clubs) and of which 4 are of each denomination (A, 2 to 10, J, Q, K). The face cards are the jacks J, queens Q and kings K.

Formula used:

Definition of combination:

  ${}_n{C}_r=\left(\begin{array}{l}n\\ r\end{array}\right)=\frac{n!}{r!\left(n-r\right)!}$  ...... (1)

With n! = n·(n-1)·.............2·1.

  $\mathrm{P}=\frac{E}{S}$  ...... (2)

Here, E is the favourable outcomes and S is the total possible outcomes.

Calculation:

Choose 5 cards, Out of the 52 cards in the normal deck.

Substitute 5 for $r$ and 52 for $n$ in equation (1)

  $\begin{array}{c}{}_{52}{C}_5=\frac{52!}{5!\left(52-5\right)!}\\ =2,598,960\left(\text{total}\text{possible}\text{outcomes}\right)\end{array}$

When all selected cards are hearts, then select 5 of the 13 hearts and thus there are C (13,5) favorable outcomes.

Substitute 5 for $r$ and 13 for $n$ in equation (1)

  $\begin{array}{c}{}_{13}{C}_5=\frac{13!}{5!\left(13-5\right)!}\\ =1287\left(\text{favourable}\text{outcomes}\right)\end{array}$

Substitute 1287 for $E$ and 2,598,960 for $S$ in equation (2)

  $\begin{array}{c}\mathrm{P}(Allhearts)=\frac{1287}{2,598,960}\\ =\frac{33}{66,640}\\ \approx 0.0005\\ =0.05\%\end{array}$

Hence, the probability that a 5-card poker hand will contain all hearts.is 0.05%.