Question

The given table shows the marks obtained by 50 students out of 100 marks in an examination:

Marks Obtained	0-25	25-50	50-75	75-100
No. of Students	10	7	20	13

(i) Find the probability that a student obtained 75 marks or above.
(ii) If $50\%$ are the passing marks, find the probability of the students failing in the exam.

Answer 1

Here, we need to find the probability that a student obtained 75 marks or above and the probability of the students failing in the exam. Using the table, we will obtain the number of students getting 75 marks or above. We will use the formula for probability to get the required probability. The number of students getting less than 50 marks is equal to the sum of the number of students getting 0-25 marks and 25-50 marks.

Formula Used: We will use the formula of the probability of an event , $P\left( E \right) = \dfrac{{{\rm{Number\, of\, favourable\, outcomes}}}}{{{\rm{Number\, of\, total\, outcomes}}}}$.
Complete step-by-step answer:
(i)
First, we will find the number of favourable outcomes and total outcomes.
The total number of students is 50. Therefore, the total number of outcomes is 50.
Next, we will use the table to find the number of students who got 75 marks or above.
We can observe that 13 students obtained 75 to 100 marks.
Therefore, the number of favourable outcomes is 13.
Finally, we will use the formula for probability of an event to calculate the probability that a student obtained marks 75 or above.
We know that the probability of an event $E$ is given by $P\left( E \right) = \dfrac{{{\rm{Number\, of\, favourable\, outcomes\,}}}}{{{\rm{Number\, of\, total\, outcomes}}}}$.
Let $E$ be the event that a student obtained 75 marks or above.
Substituting 13 for the number of favourable outcomes, and 50 for the number of total outcomes in the formula, we get
$\Rightarrow P\left( E \right) = \dfrac{{13}}{{50}}$
Therefore, the probability that a student obtained 75 marks or above is $\dfrac{{13}}{{50}}$, or $0.26$.
(ii)
First, we will find the passing marks.
It is given that $50\%$ marks of the total are passing marks.
Therefore, we get
Passing marks of the exam $= 50\% \times 100$
Simplifying the expression, we get
Passing marks of the exam $= \dfrac{{50}}{{100}} \times 100 = 50$ marks
Thus, any student who gets at least 50 marks passes the exam.
This means that all the students who get less than 50 marks fail the exam.
Let $F$ be the event that a student obtained 50 marks or less.
The total number of students is 50.
Therefore, the total number of outcomes is 50.
Next, we will use the table to find the number of students who got 50 marks or less.
We can observe that 10 students obtained 0 to 25 marks and 7 students obtained 25 to 50 marks.
Therefore, the number of students who got less than 50 marks is $10 + 7 = 17$students.
Therefore, the number of favourable outcomes is 17.
Substituting 17 for the number of favourable outcomes, and 50 for the number of total outcomes in the formula for probability of an event, we get
$\Rightarrow P\left( F \right) = \dfrac{{17}}{{50}}$
Thus, the probability that a student obtained 50 marks or less is $\dfrac{{17}}{{50}}$, or $0.34$.
Therefore, if $50\%$ are the passing marks, then the probability of the students failing in the exam is $\dfrac{{17}}{{50}}$, or $0.34$.

Note: Probability is the certainty that an event occurs. Also, we will keep in mind that in the given class intervals, the lower limit is included but the upper limit is not included. This means that all the students who got 25 marks or above, but less than 50 marks (not equal to 50 marks) come in the interval 25-50. This is why we can add the number of students in the class intervals 0-25 and 25-50 to get the number of students who got less than 50 marks.