# Two balls are drawn from an urn containing 2 white, 3 red and 4 black balls one by one without replacement. What is the probability that at least one ball is red?

Toolbox:
• $$P(A\cap\;B)=P(A)\;P(B)$$
• P (A $\cap$ B) = P(A) + P(B) - P(A $\cup$ B)
• $$\;P(B/A)=\large \frac{P(A\cap\:B)}{P(A)}$$$$\:;P(A/B)=\large \frac{P(A\cap\;B)}{P(B)}$$
Step 1:
Let $A$ be the event of not getting a red ball in first draw and $B$ be the event of not getting a red ball in second draw,
Then the required probability=Probability that at least one ball should be red.
$\Rightarrow$ 1-Probability that none is red.
$\Rightarrow 1-P(A\; and\; B)$
$\Rightarrow 1-P(A\cap B)$
$\Rightarrow 1-P(A)P( B/A)$
Step 2:
Let $P(A)$ =Probability of not getting a red ball in first draw.
(i.e)$P(A)$=Probability of getting another colour(white or black) ball in first draw.
$\Rightarrow P(A)=\large\frac{6}{9}=\frac{2}{3}$
Now that are 5 other color balls (white or black) and 3 red balls.
The other color ball can be drawn in $5C_1$ ways
$\therefore P(B/A)=\large\frac{5}{8}$
Step 3:
Hence the required probability is $1-P(A)P(B/A)$
$\Rightarrow 1-\large\frac{2}{3}\times \frac{5}{8}$
$\Rightarrow \large\frac{7}{12}$