Question

A teacher wanted to analyze the performance of two sections of students in a mathematics test of 100 marks. Looking at their performance, she found that a few students got under 20 marks and a few got 70 marks or above. So she decided to group them into the interval of varying sizes as follows: 0-20, 20-30...., 60-70, 70-100. Then she formed the following table.

Marks	Number of Students
0 – 20	7
20 – 30	10
30 – 40	10
40 – 50	20
50 – 60	20
60 – 70	15
70 – Above	8
Total	90

Find the probability that a student obtained less than 20% in the mathematics test.
Find the probability that a student obtained marks 60 or above.

Answer 1

Hint: To find out the probability of a desired outcome we are provided with the formulae
P(A) =n(E)/n(S)
Where,
P(A) = Probability of an event
n(E) = Number of desired outcome
n(S) = Total number of outcomes


Complete step by step explanation:
Case (i): Students who obtained marks less than 20%
1. Total number of students = 90
i.e. n(S), which is total number of outcomes = 90
2. Students who obtained marks less than 20% in the mathematics test = 7
i.e. n(E), which is the desired outcome = 7
3. P(A) is the probability of students who obtained marks less than 20%
Using formulae P(A) = n(E)/n(S)
And putting the values of n(E) and n(S) from step 1 and step 2
P(A) = $\dfrac{7}{{90}}$
P(A), Probability of students who got marks less than 20% = $\dfrac{7}{{90}}$
Case (ii): Students who obtained marks 60 or above.
4. Total number of students = 90
It means n(S), which is total number of outcomes = 90
5. Students obtained marks 60 or above
= Students got marks in 60-70 + Students got marks above 70
= 15 + 8
= 23
i.e. n(E), which is desired outcome = 23
6. P(A) is the probability of students who obtained marks 60 or above
Using formulae P(A) = n(E)/n(S)
And putting the values of n(E) and n(S) from step 4 and step 5
P(A) = $\dfrac{{23}}{{90}}$
P(A), Probability of students who got marks 60 or above = $\dfrac{{23}}{{90}}$


Note: 1. Care must be taken that when it is asked to find out the probability of students who have got marks less than 20%, then the number of students who fall under the interval of 20% and less should be considered not just the 20%.
2. Similarly when it is asked to find the probability of students who got marks more than 60, then all the intervals which are having a number of students who got marks more than 60 should be considered not just the 60 – 70 interval.