Q.3.3 - A First Course in Probability

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. 

Step-by-Step Solution

Answer

The conditional probability is $0.339$

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,

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

$=0.339$