Question

Let n positive integers taken randomly & multiplied together.Let $A_1$, $A_2$, $A_3$, $A_4$,$A_5$, $A_6$, $A_7$, be the events define as
$A_1$ = last digit of the product is not the number 1,3,7 or 9
$A_2$ = last digit of the product is among the number 1,3,5,7 or 9,
$A_3$ = last digit of the product is among the number 1, 3, 7 or 9
$A_4$ = last digit of lhe product 'is among the number 2, 4, 6 or 8.
$A_5$ = last digit of the product is the number 5.
$A_6$ = last digit of the product is the number 0
$A_7$ = last digit of the product is 1,2,3,4,6,7,8 or 9.On the basis of above information answer the following questions.

What is the probability that last digit of the product is $A_3$?

• A. ${\left(\frac{4}{5}\right)}^n$
• B. ${\left(\frac{2}{5}\right)}^n$
• C. ${\left(\frac{3}{5}\right)}^n$
• D. ${\left(\frac{1}{2}\right)}^n$

Answer 1

The correct option is B ${\left(\frac{2}{5}\right)}^n$

$A_3=$ last digit in the product $x=a_1,a_2.....a_a$ is among the numbers $1,3,7$ or $9$
$\therefore n\left(A_3\right)=4^{\prime \prime }$

$\therefore n\left(S\right)={10}^{\prime \prime }$
$\therefore P\left(A_3\right)=\frac{n\left(A_3\right)}{n\left(S\right)}={\left(\frac{2}{5}\right)}^{\prime \prime }$