In an examination class,60 candidates pass Integrated science or Mathematics.If 15 passed both subject and 9 more passed Mathematics, than Integrated science. find
(i) number of candidate who passed in each subject.
(ii) Probability that a candidate passed exactly one subject
If x passed IS, then
x+9 + x - 15 = 60
(i) solve for x, then you can answer this
(b) 15 passed both, so 45/60 passed exactly one.
Let's solve this step-by-step:
(i) Number of candidates who passed in each subject:
Let's assume the number of candidates who passed Integrated Science is x, and the number of candidates who passed Mathematics is y.
From the given information, we know that 60 candidates passed either Integrated Science or Mathematics.
According to the principle of inclusion-exclusion,
Total = Passed in Integrated Science + Passed in Mathematics - Passed in both subjects
Therefore, we can write the equation as:
x + y - 15 = 60 ----(1)
We are also given that 9 more candidates passed Mathematics than Integrated Science. Mathematically, we can write this as:
y = x + 9 ----(2)
To find the solution, we can substitute the value of y from equation (2) into equation (1):
x + (x + 9) - 15 = 60
2x - 6 = 60
2x = 60 + 6
2x = 66
x = 66/2
x = 33
Now that we have the value of x, we can substitute it back into equation (2):
y = x + 9
y = 33 + 9
y = 42
So, the number of candidates who passed Integrated Science is 33, and the number of candidates who passed Mathematics is 42.
(ii) Probability that a candidate passed exactly one subject:
The probability of an event happening is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.
In this case, the number of favorable outcomes is the number of candidates who passed in exactly one subject, which is the sum of candidates who passed Integrated Science only (33 - 15 = 18) and the sum of candidates who passed Mathematics only (42 - 15 = 27). Therefore, the number of favorable outcomes is 18 + 27 = 45.
The total number of possible outcomes is the total number of candidates, which is 60.
Hence, the probability that a candidate passed exactly one subject is:
45/60 = 3/4 = 0.75 (or 75%)
So, the probability is 0.75 or 75%.
To find the number of candidates who passed in each subject, we can use Venn diagrams.
Let's assume that the number of candidates who passed Integrated Science is represented by "IS" and the number of candidates who passed Mathematics is represented by "M". The number of candidates who passed both subjects is represented by "B".
From the given information, we know that 60 candidates passed Integrated Science or Mathematics. So, we have:
IS + M - B = 60 ----- (1)
We are also given that 15 candidates passed both subjects. So, we have:
B = 15 ----- (2)
Additionally, we know that 9 more candidates passed Mathematics than Integrated Science. So, we have:
M = IS + 9 ----- (3)
To solve these equations, we can substitute equation (2) into equations (1) and (3):
IS + IS + 9 - 15 = 60 (from equation (1) and (2))
2IS - 6 = 60
2IS = 66
IS = 33
Substituting this value of IS into equation (3):
M = 33 + 9
M = 42
Therefore, the number of candidates who passed Integrated Science is 33, and the number of candidates who passed Mathematics is 42.
Now, let's move on to finding the probability that a candidate passed exactly one subject.
To find this probability, we need to calculate the number of candidates who passed exactly one subject and divide it by the total number of candidates.
The number of candidates who passed exactly one subject is the sum of the candidates who passed Integrated Science only and the candidates who passed Mathematics only.
From our previous calculations, we know that the number of candidates who passed Integrated Science only is 33 - 15 = 18, and the number of candidates who passed Mathematics only is 42 - 15 = 27.
Therefore, the number of candidates who passed exactly one subject is 18 + 27 = 45.
The total number of candidates is given as 60.
So, the probability that a candidate passed exactly one subject is:
P(Exactly one subject) = Number of candidates who passed exactly one subject / Total number of candidates = 45 / 60 = 0.75 (or 75%)
Hence, the probability that a candidate passed exactly one subject is 0.75 or 75%.