Question

Numbers are selected at random, one at a time, from the two-digit numbers 00, 01, 02,..., 99 with replacement. An event E occurs if and only if the product of the two digits of a selected number is 18. If four numbers are selected, find probability that the event E occurs at least 3 times.

• A. $\frac{1}{{25}^4}$
• B. $\frac{96}{{25}^4}$
• C. $\frac{97}{{25}^4}$
• D. $\frac{95}{{25}^4}$

Answer 1

The correct option is C $\frac{97}{{25}^4}$

Let E be the event that product of the two digits is 18, therefore required numbers are 29, 36, 63 and 92.
Hence, $p=P\left(E\right)=\frac{4}{100}$
and probability of non - occurrence of E is
$q=1-P\left(E\right)=1-\frac{4}{100}=\frac{96}{100}$
Out of the four numbers selected, the probability that the event E occurs atleast 3 times, is given as
$P{=}^4{C}_3{p}^{3}q{+}^4{C}_4{p}^4=4{\left(\frac{4}{100}\right)}^3\left(\frac{96}{100}\right)+{\left(\frac{4}{100}\right)}^4=\frac{97}{{25}^4}$