In a class of 75 students,15 are above average,45 are average and the rest below average achievers.The probability that an above average achieving student fails is 0.005,then an average achieving student fails is 0.05 and the probability of a below average achieving student failing is 0.15.If a student is known to have passed,what is the probability that he is a below average achiever?

Question

In a class of 75 students,15 are above average,45 are average and the rest below average achievers.The probability that an above average achieving student fails is 0.005,then an average achieving student fails is 0.05 and the probability of a below average achieving student failing is 0.15.If a student is known to have passed,what is the probability that he is a below average achiever?

Answer 1

let O be over average, A be average, and B be below average, F for fail and P for pass

prob(O) = 15/75 = 1/5
prob(A) = 45/75 = 3/5
prob(B) = 15/45 = 1/5
prob(P) = x
prob(F) = 1-x , not really needed

so we have
AF = .05
AP = (3/5)x
OF = .005
OP = (1/5)x
BF = .15
BP = (1/5)x

.05 + 3x/5 + .005 + x/5 + .15 + x/5= 1
x = 1 - .05 - .005 - .15 = .795

Thus:
AF = .05
AP = .477
OF = .005
OP = .159
BF = .15
BP = .159 , notice they add up to 1

So we have BP = .159
B(.795) = .159
B = .159/.795 = .2

Answer 2

To find the probability that a student who has passed is a below average achiever, we can use Bayes' theorem. Bayes' theorem states that:

P(A|B) = (P(B|A) * P(A)) / P(B)

Where:
P(A|B) is the probability of event A occurring given that event B has occurred.
P(B|A) is the probability of event B occurring given that event A has occurred.
P(A) is the probability of event A occurring.
P(B) is the probability of event B occurring.

Let's calculate each component step by step.

Step 1: Calculate the probability of passing for each achievement level.
The probability of passing for an above-average achiever is 1 - 0.005 = 0.995.
The probability of passing for an average achiever is 1 - 0.05 = 0.95.
The probability of passing for a below-average achiever is 1 - 0.15 = 0.85.

Step 2: Calculate the probability of being each achievement level.
The probability of being an above-average achiever is 15/75 = 1/5.
The probability of being an average achiever is 45/75 = 3/5.
The probability of being a below-average achiever is (75 - 15 - 45)/75 = 15/75 = 1/5.

Step 3: Calculate the probability of passing.
The probability of passing can be calculated by considering the different achievement levels:
P(Pass) = (P(Above average) * P(Pass|Above average)) + (P(Average) * P(Pass|Average)) + (P(Below average) * P(Pass|Below average))
= (1/5 * 0.995) + (3/5 * 0.95) + (1/5 * 1)
= 0.199 + 0.57 + 0.2
= 0.969

Step 4: Calculate the probability of being a below-average achiever given that a student has passed.
P(Below average|Pass) = (P(Pass|Below average) * P(Below average)) / P(Pass)
= (1/5 * 1) / 0.969
= 1/5 / 0.969
≈ 0.206

Therefore, the probability that a student who has passed is a below-average achiever is approximately 0.206, or 20.6%.

Answer 3

To solve this problem, we can use Bayes' theorem.

Let's define the events:
A: Above average achiever
B: Average achiever
C: Below average achiever
P: Passed

We are required to find P(C | P), which means the probability that the student is a below average achiever given that the student has passed (P(P)).

We can calculate this using Bayes' theorem:

P(C | P) = (P(P | C) * P(C)) / P(P)

We know:
P(P | C) = probability of passing given the student is a below average achiever = 1 - probability of failing for a below average achiever = 1 - 0.15 = 0.85
P(C) = probability of being a below average achiever = (total below average students)/(total number of students) = (75 - 15 - 45)/75 = 15/75 = 1/5
P(P) = probability of passing = 1 - probability of failing

To calculate P(P), we need to find the weighted average of failing probabilities for all the categories:

P(P | A) = 1 - probability of failing for an above average achiever = 1 - 0.005 = 0.995
P(P | B) = 1 - probability of failing for an average achiever = 1 - 0.05 = 0.95
P(P | C) = 1 - probability of failing for a below average achiever = 1 - 0.15 = 0.85

Now, P(P) can be calculated as:

P(P) = (P(P | A) * P(A)) + (P(P | B) * P(B)) + (P(P | C) * P(C))

P(A) = probability of being an above average achiever = 15/75 = 1/5
P(B) = probability of being an average achiever = 45/75 = 3/5

Substituting these values in:

P(P) = (0.995 * 1/5) + (0.95 * 3/5) + (0.85 * 1/5)

Now, let's calculate the final probability:

P(C | P) = (P(P | C) * P(C)) / P(P)

= (0.85 * 1/5) / [ (0.995 * 1/5) + (0.95 * 3/5) + (0.85 * 1/5) ]

Simplifying this expression will give us the final probability that a student who has passed is a below average achiever.