Question

Two decks of cards are there. Each deck contains $20$ cards, with numbers from $1$ to $20$written on them. A card is drawn of random from each deck, getting the numbers $x$ and $y$ what is the probability that $\log x+\log y$ is a positive integer. Logs are taken to the base $10$.

Answer 1

Step 1: Finding the sample space:

Sample space for drawing two cards from each deck is

$$\begin{gathered} \left\{\begin{array}{l}\left.\begin{array}{r}(1,1),(1,2),(1,3),(1,4),(1,5),.................,(1,20), \\ (2,1),(2,2),(2,3),(2,4),(2,5),.................,(2,20), \\ ..................................................................... \\ ...................................................................... \\ \\ ......................................................................\end{array}\right\}\\ (20,1),(20,2),(20,3),(20,4),(20,5),............(20,20)\end{array}\right. \end{gathered}$$

Total number of outcomes $=400$

Step 2: Finding the favorable outcomes:

Since, $\log x+\log y=\log \left(xy\right)$

${\log }_{10}\left(xy\right)$ is a positive integer if $xy=10,100,1000$.

$i.e.{\log }_{10}\left(10\right)=1;{\log }_{10}\left(100\right)=2;{\log }_{10}\left(1000\right)=3$

$\Rightarrow$ $\log xy$ is a positive integer when $(x,y)=(1,10),(10,1),(10,10),(5,20),(20,5),(2,5),(5,2)$

$\therefore$Number of favorable outcomes $=7$

Step 3: Finding the probability:

We know, Probability of an event $=\frac{Numberoffavorableoutcomes}{Totalnumberofoutcomes}$

The required probability $=\frac{7}{400}$

Thus, the probability that $\log x+\log y$ is a positive integer is $\frac{7}{400}$.