Question

Your friend writes a phone number down on a piece of paper but the last three numbers or gets much after being in your pocket all day long you decide to randomly choose numbers for each of the three digit use a sample space to determine whether guessing the first digit correctly in the second digit correctly are independent events

Answer 1

Repetition of 10 digits is allowed. So, the number of possible outcomes in the sample space:

$10\cdot10=100$

Events A and B are independent if

$P(A\ and\ B)=P(A)\cdot P(B)$

From the sample space:

$P(first\ correct)=P(second\ correct)=1/10$

Then:

$P(first\ correct)\cdot P(second\ correct)=1/100$

$P(first\ correct\ and\ second \ correct)=1/100$

Since $P(first\ correct)\cdot P(second\ correct)=P(first\ correct\ and\ second \ correct)$ ,

the events are independent.