Question

Out of 20 cards numbered from 1 to 20, which are mixed thoroughly, a card is drawn at random. Find the probability that the drawn card bears a number which is a multiple of 3 or 7.

Answer 1

In this question, we are given 20 cards numbered 1 to 20. We have to find the probability of getting a number which is a multiple of 3 or 7 when a card is drawn at random. For this, we will first find the probability of getting a number which is multiple of 3 and then find the probability of getting a number a multiple of 7. After that, we will add them to get required probability. For finding probability we will use the formula: $\text{Probability}=\dfrac{\text{Number of favorable outcomes}}{\text{Total outcomes}}$. We will find a number of favorable outcomes by finding numbers which are multiple of 3 between 1 to 20 and numbers which are multiple of 7 between 1 to 20.

Complete step-by-step answer:
Here, we are given 20 cards numbered from 1 to 20.
We need to find the probability of getting a multiple of 3 or 7, when a card is drawn out randomly.
Numbers on the card are:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20.
Now, let us find the probability of getting a multiple of 3. Multiple of 3 between 1 to 20 are 3, 6, 9, 12, 15, 18. As we can see there are 6 numbers which are multiple of 3, so the number of favorable outcomes are 6. There are a total of 20 numbers. Since,
$\text{Probability}=\dfrac{\text{Number of favorable outcomes}}{\text{Total outcomes}}$.
Therefore, probability of getting a multiple of 3 is given by $\dfrac{6}{20}$.
Now, let us find the probability of getting a multiple of 7. Multiple of 7 between 1 to 20 are 7, 14. As we can see, there are 2 numbers which are multiple of 7, so the number of favorable outcomes are 2. There are a total of 20 numbers. Therefore, probability of getting a multiple of 7 is given by $\dfrac{2}{20}$.
Since, we need to find the probability of getting a multiple of 3 or 7, so we get probability as,
$\dfrac{6}{20}+\dfrac{2}{20}=\dfrac{8}{20}$.
Now, $\dfrac{8}{20}$ can be simplified as $\dfrac{2}{5}$.
So, $\dfrac{2}{5}$ is our required probability.

Note: Students should note that we have added both the probabilities since we needed multiple of 3 or multiple of 7. We can also solve it by directly counting multiple of 3 or 7 between 1 to 20 which will be our favorable outcome (3, 6, 7, 9, 12, 14, 15, 18). Favorable outcomes will be 8 and total outcomes will be 20. So, probability will be $\dfrac{8}{20}=\dfrac{2}{5}$. Students should be careful while counting the favorable outcome.