Question

A number is chosen at random among the first \[120\] natural numbers. The probability of the number chosen to be a multiple of \[5\]or \[15\] is
1) \[\dfrac{1}{5}\]
2) \[\dfrac{1}{8}\]
3) \[\dfrac{1}{24}\]
4) \[\dfrac{1}{6}\]

Answer 1

Hint:In this type of question we can observe the number of outcomes of the asked probability if we find that we can easily put it in the basic formula of probability and that is
\[\Pr obailty=\dfrac{Number\_of\_possible\_outcomes}{Total\_number\_of\_outcomes}\] and we will get the required probability.
Complete step-by-step solution:
So first we must know what the probability is and its importance and mathematical formula for finding the probability.
Probability means possibility. It is a branch of mathematics that deals with the occurrence of a random event. The value is expressed from zero to one. Probability has been introduced in Math to predict how likely events are to happen.
The meaning of probability is basically the extent to which something is likely to happen. This is the basic probability theory, which is also used in the probability distribution, where you will learn the possibility of outcomes for a random experiment. To find the probability of a single event to occur, first, we should know the total number of possible outcomes.
Probability Definition in Math:
Probability is a measure of the likelihood of an event to occur. Many events cannot be predicted with total certainty. We can predict only the chance of an event to occur i.e. how likely they are to happen, using it. Probability can range from \[0\] to \[1\] , where \[0\] means the event to be an impossible one and \[1\] indicates a certain event. Probability for students is an important topic which explains all the basic concepts of this topic. The probability of all the events in a sample space adds up to 1.
For example, when we toss a coin, either we get Head or Tail, only two possible outcomes are possible \[(H,T)\] . But if we toss two coins in the air, there could be three possibilities of events to occur, such as both the coins show heads or both show tails or one shows heads and one tail, i.e.\[(H,H),(H,T),(T,T)\].
So now coming to the question,
We need to find the possible outcomes among first \[120\] natural numbers that must be divisible by \[5\] or \[15\].
Let \[A\] be the set of possible outcomes that are divisible by \[5\] .
Therefore,
\[A=\{5,10,15,...,115,120\}\]
That is we have total \[24\] possible outcomes.
Now let \[B\] be the set of possible outcomes that are divisible by \[15\] .
Therefore,
\[B=\{15,30,45,...,105,120\}\]
That is we have total \[8\] possible outcomes.
So if we notice \[B\subset A\] that is \[B\] is a subset of \[A\] therefore,
\[A\cup B=A\]
So now we need to find the probability of a number among first \[120\] natural numbers that must be divisible of \[5\] or \[15\].
Therefore,
\[P(A\cup B)=\dfrac{Number\_of\_possible\_outcomes}{Total\_number\_of\_outcomes}\]
Since we know \[A\cup B=A\] ,
Therefore,
\[\begin{align}
  & \Rightarrow P(A)=\dfrac{24}{120} \\
 & \Rightarrow P(A)=\dfrac{1}{5} \\
\end{align}\]
Therefore our final answer is option \[(1)\]

Note:One practical use for probability distributions and scenario analysis in business is to predict future levels of sales. It is essentially impossible to predict the precise value of a future sales level; however, businesses still need to be able to plan for future events.