IN A CLASS OF 130 STUDENTS.85 PASSED IN MATHS, 60 PASSED IN SCIENCE, 10 FAILED IN BOTH.
a) how many passed in science but failed in math
b)how many failed in exactly one subject
If 10 failed in both, that means 130-10=120 passed at least one subject.
Since 85 passed math, 60 passed science, there are 145 passes from 120 students. We conclude that 145-120=25 passed both.
Hence (b) 120-25=95 passed (or failed) exactly one subject.
For (a), 60 passed science, and 25 passed both, so 35 passed science but did not make math.
A more mathematical treatment would be:
P'(A∪B)=1-P(A∪B)
P(A)=85/130
P(B)=60/130
P'(A∪B)=1-P(A∪B)=10/130
=>
P(A∪B)=(130-10)/130=120/130
We also know that:
P(A∪B)=P(A)+P(B)-P(A∩B)
120/130=85/130+60/130-P(A∩B)
=>
P(A∩B)=85/130+60/130-120/130=25/130
The cardinality of a set is the number of members in the set.
For example, the cardinality of A is denoted by |A|, which is equal to the number of students who passed math=85.
Hence |B|=60, |A∩B|=25, |A∪B|=120.
(a)
|A'∩B|=|B|-|A∩B|=60-25=35
(b)
|A∪B|-|A∩B|=120-25=95
a) Well, it seems like science is a bit more challenging than math. There were 60 students who passed in science, but we know that 10 students failed in both subjects. To find out the number of students who passed in science but failed in math, we need to subtract the number of students who failed in both (10) from the total number of students who passed in science (60). So, 60 - 10 = 50 students passed in science but failed in math.
b) Ah, the classic case of "almost there, but not quite." To determine how many students failed in exactly one subject, we can add the number of students who failed in math but passed in science with the number of students who passed in math but failed in science. Let's calculate it. Since there were 85 students who passed in math and 60 students who passed in science, we can subtract the number of students who passed in both subjects (10) from both of those totals and add them up. So, 85 - 10 + 60 - 10 = 125 students failed in exactly one subject.
To solve this problem, we can use a Venn diagram to visualize the information given.
Let's denote:
M = Mathematics
S = Science
F = Failed in both subjects
a) To find the number of students who passed in Science but failed in Math, we need to calculate the intersection of the Science set (S) and the complement of the Math set (M').
From the given information:
Total students = 130
Passed in Math (M) = 85
Passed in Science (S) = 60
Failed in both (F) = 10
To find the number of students who passed in Science but failed in Math, we can use the formula:
n(S ∩ M') = n(S) - n(F)
n(S ∩ M') = 60 - 10
n(S ∩ M') = 50
Therefore, 50 students passed in Science but failed in Math.
b) To find the number of students who failed in exactly one subject, we can add the number of students who failed in Math only (M - F) and the number of students who failed in Science only (S - F).
n(M - F) = n(M) - n(F) = 85 - 10 = 75
n(S - F) = n(S) - n(F) = 60 - 10 = 50
n(M - F ∪ S - F) = n(M - F) + n(S - F)
n(M - F ∪ S - F) = 75 + 50
n(M - F ∪ S - F) = 125
Therefore, 125 students failed in exactly one subject.
To find the answers to these questions, we'll use the concept of set theory and Venn diagrams. Let's break down the problem step by step.
a) How many passed in science but failed in math?
To find the number of students who passed in science but failed in math, we need to calculate the intersection of the set of students who passed in science and the set of students who failed in math.
1. Start with the total number of students who passed in science: 60.
2. Subtract the number of students who failed in both subjects: 10.
So, the total number of students who passed in science but failed in math is 60 - 10 = 50.
b) How many failed in exactly one subject?
To find the number of students who failed in exactly one subject, we need to calculate the union of the sets of students who failed in math and the students who failed in science, and then subtract the number of students who failed in both subjects.
1. Start with the total number of students who failed in math: 130 - 85 = 45.
2. Add the total number of students who failed in science: 60.
3. Subtract the number of students who failed in both subjects: 10.
So, the total number of students who failed in exactly one subject is 45 + 60 - 10 = 95.