Question

An urn contains $5$ white balls and $3$ red balls. A sample of $5$ balls is selected at random from the urn. What is the probability that sample contains $3$ white balls and $2$ red balls?

• A. $\frac{15}{28}$
• B. $\frac{9}{28}$
• C. $\frac{27}{28}$
• D. $\frac{21}{28}$

Answer 1

The correct option is A $\frac{15}{28}$

Given:
Total number of white balls $=5$
Total number of red balls $=3$

Step 1: $3$ white balls can be selected by ${}^5{\text{C}}_3$ ways
$\Rightarrow {}^5{\text{C}}_3=10$

Step 2: $2$ red balls can be selected by ${}^3{\text{C}}_2$ ways
$\Rightarrow {}^3{\text{C}}_2=3$

Step 3: $5$ balls can be selected for the sample by ${}^8{\text{C}}_5$ ways
$\Rightarrow {}^8{\text{C}}_5={}^8{\text{C}}_3$
$=\frac{8\times 7\times 6}{3\times 2}$
$=56$

Step 4: Probability of selecting $3$ white balls and $2$ red balls for the sample
$=\frac{{}^5{\text{C}}_3\times {}^3{\text{C}}_2}{{}^8{\text{C}}_5}$
$=\frac{10\times 3}{56}$
$=\frac{15}{28}$