Question

In a hand of bridge, find the probability that you have$5$spades and your partner has the remaining$8$.

Short answer

$0.0000175\dots$

The outcome space is all possible hands of the partner.

Step-by-step solution

Given Information.

Given that in the hand of the bridge.

Explanation.

Experiment: A fair game of bridge. You have $5$spades.

$WhatistheprobabilityofA=\{yourpartnerhas9spades\}$

The outcome space $S$contains all the possible hands of my partner. Each of the $\left(\begin{array}{l}39\\ 13\end{array}\right)$elements $S$has an equal probability of happening (in a fair game).

Now if all of the $9$spades have to be in the hand, the remaining$5$cards can be chosen from the$30$in $\left(\begin{array}{c}30\\ 5\end{array}\right)$ways. This is the number of hands in the event$A$.

The Axioms of probability for a sample space of equally likely events derive the formula:

$P\left(A\right)=\frac{\left|A\right|}{\left|S\right|}A\subseteq S,\left|X\right|-\text{the number of elements of}X$

Finally:

$P\left(A\right)=\frac{\left(\begin{array}{c}30\\ 5\end{array}\right)}{\left(\begin{array}{l}39\\ 13\end{array}\right)}=\frac{13}{740962}=0.00001754475\dots$