Question

Five unbiased cubical dice are rolled simultaneously. Let $m$ and $n$ be the smallest and the largest number appearing on the faces of the dice, then match the following event given in column I with their respective probabilities given in Column II.

Answer 1

$A)$ For $m=3$(smallest)

So, dice can show $3,4,5,6$

Total no. of outcome $=6^5$

Probability $={\left(\frac{4}{6}\right)}^5={\left(\frac{2}{3}\right)}^5$

$B)$ $n=4$ then no. of favourable outcomes $=1,2,3,4$

Probability $={\left(\frac{4}{6}\right)}^5={\left(\frac{2}{3}\right)}^5$

$C)2\le m\le 4$

If $m=2$ then favourable outcome $=2,3,4,5,6$

$P={\left(\frac{5}{6}\right)}^5$

If $m=3$ then favourable outcome $=3,4,5,6$

$P={\left(\frac{2}{3}\right)}^5$

If $m=4$ then favourable outcome $=4,5,6$

$P={\left(\frac{3}{6}\right)}^5={\left(\frac{1}{2}\right)}^5$

Total probability$={\left(\frac{5}{6}\right)}^5+{\left(\frac{2}{3}\right)}^5+{\left(\frac{1}{2}\right)}^5$

$D)$ $m=2$ and $n=5$

For $m=2$ favourable outcome $=2,3,4,5,6$

$P={\left(\frac{5}{6}\right)}^5$

For $n=5,$ favourable outcome $=1,2,3,4,5$

Probability $={\left(\frac{5}{6}\right)}^5$

Total Probability $=2{\left(\frac{5}{6}\right)}^5$