Out of 100 cards with numbers 00; 01; 02; ... ... .; 98; 99 is chosen at random. Let X and Y, respectively, be the sum and product of the numbers on the selected card. Find the conditional probability of an event P (X = i | Y = 0), i = 0; 1; 2; ... ... ... 18;
Can you, please, explain why i changes in this interval i = 0; 1; 2; ... ... ... 18;, if we have 100 cards and, in general, what is the point of finding the conditional probability of the sum and product of the number on the card?
Let's start with $Y=0$. This means at least one of the digits shown on the card is zero. That is the only way to achieve a product of 0. So, for the conditional probability, your sample space changes to the following cards:
$$00, 01, 02, 03, 04, 05, 06, 07, 08, 09, 10, 20, 30, 40, 50, 60, 70, 80, 90$$
Now, let's break it down:
$$P(X=i|Y=0) = \begin{cases} \tfrac{1}{19}, & i=0 \\ \tfrac{2}{19}, & 1\le i \le 9 \\ 0, & 9 < i\end{cases}$$
It is not possible to get a sum of 10 or more when one of the digits shown must be zero. For other products, you similarly limit your sample space. Then, from that limited sample space, you can calculate how many cards yield each possible sum.