# A number is selected at random from the first 50 natural numbers. Find the probability that it is a multiple of 3 and 4.

Last updated date: 20th Mar 2023

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Hint: First of all find the number of multiples of 3 x 4 = 12 from 1 to 50. Then find the probability of getting a multiple of 3 and 4 by using the formula, \[\dfrac{\text{Number of favorable outcomes}}{\text{Number of total outcomes}}\].

We are given that a number is selected at random from the first 50 natural numbers. Here, we have to find the probability that it is a multiple of 3 and 4.

Before proceeding with this question, we must know what a natural number is. Natural numbers are a part of the number system which includes all the positive integers from 1 till infinity. It should be noted that natural numbers include only positive integers.

Also, the term probability means the extent to which an event is likely to occur measured by ratio of the favorable cases to the whole numbers of cases possible. In short, we can say that ‘probability’ means possibility.

First of all, we will find the multiples of 3 from 1 to 50, that are,

\[\text{3, 6, 9, }\underline{\text{12}}\text{, 15, 18, 21, }\underline{\text{24}}\text{, 27, 30, 33, }\underline{\text{36}}\text{, 39, 42, 45, }\underline{\text{48}}\]

Now, we will find the multiples of 4 from 1 to 50, that are

\[\text{4, 8, }\underline{\text{12}}\text{, 16, 20, }\underline{\text{24}}\text{, 28, 32, }\underline{\text{36}}\text{, 40, 44, }\underline{\text{48}}\]

Now in the question we are asked that, we have to find the number such that it is multiple of 3 and 4. So, we will find the multiples that are common to both 3 and 4, that are

\[\text{12, 24, 36, 48}\]

Hence, we can see that if we have to find the number that is multiple of both 3 and 4, its better to find the multiples of 3 x 4 = 12 apart from finding the multiple of 3 and 4 individually and taking the common terms from them.

Therefore, multiple of 3 and 4 or multiple of 12 from 1 to 50 are,

\[\text{12, 24, 36, 48}\]

Hence, we get the number of multiples of 3 and 4 = 4 from 1 to 50.

Now, we know the total numbers from 1 to 50 = 50.

Now, we know that,

\[\text{Probability of an event to happen, P(E)}=\dfrac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\]

Hence, we have the number of favorable outcomes that is the number of multiples of 3 and 4 from 1 to 50 = 4.

Also, we have a number of total outcomes that is total numbers from 1 to 50 = 50.

Therefore, we get the probability of getting the multiple of 3 and 4, when a number is selected at random from first 50 natural numbers

\[=\dfrac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\]

\[=\dfrac{4}{50}\]

\[=\dfrac{2}{25}\]

Note: Here, some students make this mistake of taking all the multiples of 3 and 4. But they must note that we are asked to pick the multiple of 3 and 4 that means 3 and 4 both. In case, we would be asked to pick the multiple of 3 and 4, there we would take all the multiples of 3 as well 4. Also, it is always better and more reliable to find the multiples of 3 x 4 = 12 than to find the multiples of 3 and 4 individually and taking common terms from them.

We are given that a number is selected at random from the first 50 natural numbers. Here, we have to find the probability that it is a multiple of 3 and 4.

Before proceeding with this question, we must know what a natural number is. Natural numbers are a part of the number system which includes all the positive integers from 1 till infinity. It should be noted that natural numbers include only positive integers.

Also, the term probability means the extent to which an event is likely to occur measured by ratio of the favorable cases to the whole numbers of cases possible. In short, we can say that ‘probability’ means possibility.

First of all, we will find the multiples of 3 from 1 to 50, that are,

\[\text{3, 6, 9, }\underline{\text{12}}\text{, 15, 18, 21, }\underline{\text{24}}\text{, 27, 30, 33, }\underline{\text{36}}\text{, 39, 42, 45, }\underline{\text{48}}\]

Now, we will find the multiples of 4 from 1 to 50, that are

\[\text{4, 8, }\underline{\text{12}}\text{, 16, 20, }\underline{\text{24}}\text{, 28, 32, }\underline{\text{36}}\text{, 40, 44, }\underline{\text{48}}\]

Now in the question we are asked that, we have to find the number such that it is multiple of 3 and 4. So, we will find the multiples that are common to both 3 and 4, that are

\[\text{12, 24, 36, 48}\]

Hence, we can see that if we have to find the number that is multiple of both 3 and 4, its better to find the multiples of 3 x 4 = 12 apart from finding the multiple of 3 and 4 individually and taking the common terms from them.

Therefore, multiple of 3 and 4 or multiple of 12 from 1 to 50 are,

\[\text{12, 24, 36, 48}\]

Hence, we get the number of multiples of 3 and 4 = 4 from 1 to 50.

Now, we know the total numbers from 1 to 50 = 50.

Now, we know that,

\[\text{Probability of an event to happen, P(E)}=\dfrac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\]

Hence, we have the number of favorable outcomes that is the number of multiples of 3 and 4 from 1 to 50 = 4.

Also, we have a number of total outcomes that is total numbers from 1 to 50 = 50.

Therefore, we get the probability of getting the multiple of 3 and 4, when a number is selected at random from first 50 natural numbers

\[=\dfrac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\]

\[=\dfrac{4}{50}\]

\[=\dfrac{2}{25}\]

Note: Here, some students make this mistake of taking all the multiples of 3 and 4. But they must note that we are asked to pick the multiple of 3 and 4 that means 3 and 4 both. In case, we would be asked to pick the multiple of 3 and 4, there we would take all the multiples of 3 as well 4. Also, it is always better and more reliable to find the multiples of 3 x 4 = 12 than to find the multiples of 3 and 4 individually and taking common terms from them.

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