Question

The probabilities of two students A and B coming to the school in time are \[\dfrac{3}{7}\] and \[\dfrac{5}{7}\] respectively. Assuming that the events ‘A coming in time ‘and ‘B coming in time’ are independent. Find the probability of only one of them coming to the school in time. Write at least one advantage of coming to school on time.

Answer 1

Hint: To solve this problem, we will use the given condition and apply the concept of Venn diagram and probability to get the answer. The probability of not coming in time is equal to 1 – probability of coming in time. Probability of an event can be between 0 and 1 but it cannot exceed 1 because the maximum probability of an event is equal to 1.

Complete step-by-step answer:

Now, probability that A comes school int time i.e. P (A) = \[\dfrac{3}{7}\]
Probability that B comes school in time i.e. P (B) = \[\dfrac{5}{7}\]
Now, the given condition is that both the probabilities are independent which means the probability of their intersection is equal to the product of both probabilities. Therefore,
P (${\text{A}} \cap {\text{B}}$) = P (A) x P (B).
Now, we have to find the probability when only one of them comes to school on time. We will use the Venn diagram to find the probability.

The above Venn diagram shows the probability of both event A and event B. So, from the Venn diagram, we can see that P (${\text{A}} \cap {\text{B'}}$) represents probability when A comes in time and B not in time. Similarly, P (${\text{A'}} \cap {\text{B}}$) represents probability when b comes in time while A does not in time.
Therefore, Probability when only one comes in time = P (${\text{A}} \cap {\text{B'}}$) + P (${\text{A'}} \cap {\text{B}}$)
Now, given probabilities are independent. So, P (${\text{A}} \cap {\text{B'}}$) = P (A). P (${\text{B'}}$) and P (${\text{A'}} \cap {\text{B}}$) = P (${\text{A'}}$). P (B).
So, required probability = $\dfrac{3}{7}\left( {1{\text{ - }}\dfrac{5}{7}} \right){\text{ + }}\left( {1{\text{ - }}\dfrac{3}{7}} \right)\dfrac{5}{7}$ = $\dfrac{{26}}{{49}}$
So, the probability when only one of them comes to school in time is $\dfrac{{26}}{{49}}$.
The advantage of coming to school on time is that you don’t miss any class or lecture of your subject and hence is in tune with studies.

Note: When we come up with such types of problems, the easiest way to solve the problem is that you should draw a Venn diagram. Venn diagrams help in understanding which probability we have to find and if you use a Venn diagram, the chances of incorrect answers also decreases. Use the given condition properly as questions can be solved only with the help of the given condition.