Question

Use Equation $(2.1)$to compute in a hand of bridge the conditional probability that East has $3$spades given that North and South have a combined total of $8$ spades.

Short answer

The conditional probability is$0.339$

Step-by-step solution

Step 1:Given Information

The probability that East has $3$spades given that North and South have a consolidated complete of $8$ spades is,

$\mathrm{P}(\text{East has}3\text{spades}\mid \text{North-South has}8\text{spades})=\frac{P(\text{East has}3,\mathrm{N}-\mathrm{Shas}8)}{P(\mathrm{N}-\mathrm{S}\text{has}8)}$

Step 2:Explanation

The probability that north and south has 8 spades is,

$P(\mathrm{N}-\mathrm{S}\text{has}8)=\frac{\left(\begin{array}{l}13\\ 8\end{array}\right)\left(\begin{array}{l}39\\ 18\end{array}\right)}{\left(\begin{array}{l}52\\ 26\end{array}\right)}$

$=0.1618$

The probability that East has $3$and North and South has $8$spades is

$P\left(\text{East has}3\text{and N-Shas}8\right)=\frac{\left(\begin{array}{l}13\\ 8\end{array}\right)\left(\begin{array}{l}39\\ 18\end{array}\right)\left(\begin{array}{l}5\\ 3\end{array}\right)\left(\begin{array}{l}21\\ 10\end{array}\right)}{\left(\begin{array}{l}52\\ 26\end{array}\right)\left(\begin{array}{l}26\\ 13\end{array}\right)}$

$=0.0548$

Step 3:Final Answer

The conditional probability that East has $3$spades and North and South have a combined total of $8$ spades,

$\text{P (East has}3\text{spades}\mid \text{North-South has}8\text{spades})=\frac{0.0548}{0.1618}$

$=0.339$