Question

An urn contains $10$ black and $5$ white balls. Two balls are drawn from the urn one after the other without replacement. What is the probability that both drawn balls are black?

Answer 1

Given,
Total number of balls $=15$
Number of black balls $=10$
Number of white balls $=5$
Let $A\text{and}B$ denote the events that first
and second ball drawn are black.

According to question
$P\left(A\right)$ = $P\left(\text{black ball in first draw}\right)=\frac{10}{15}$

One black ball is already drawn then,

$\text{Remaining balls}=15-1=14$
$\text{No of black balls}=10-1=9$

$P\left(B|A\right)=\frac{9}{14}$
By multiplication of probability,

$P(A\cap B)=P\left(A\right)\times P(A\cap B)$ ...$\left(1\right)$

Putting the value of $P\left(A\right)$ & $P\left(B|A\right)$ in $\left(1\right)$

$P(A\cap B)=\frac{10}{15}\times \frac{9}{14}$

$\Rightarrow P(A\cap B)=\frac{3}{7}$.