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The overall percentage of failure in a certain examination is 0.30. What is the probability that out of a group of 6 candidates at least 4 passed the examination?
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The overall percentage of failure in a certain examination is 0.30. Wh...
Solution:

Given,
Overall percentage of failure in a certain examination is 0.30.
Hence, the probability of failure in an examination is 0.30.

To find:
The probability that out of a group of 6 candidates at least 4 passed the examination.

Approach:
We will use the binomial distribution formula to solve this problem.

Binomial Distribution Formula:
P(X = x) = nCx * p^x * q^(n-x)
Where,
n = number of trials
x = number of successes
p = probability of success
q = probability of failure
nCx = n! / (x! * (n-x)!)

Let's solve this problem step by step.

Step 1:
Let's find the probability of passing an examination.
p = 1 - 0.30 (as the probability of failure is given)
p = 0.70

Step 2:
Let's find the probability of at least 4 candidates passing the examination.
P(X ≥ 4) = P(X = 4) + P(X = 5) + P(X = 6)
Where,
X = number of candidates passing the examination

P(X = x) = nCx * p^x * q^(n-x)
For x = 4,
P(X = 4) = 6C4 * 0.70^4 * 0.30^2
P(X = 4) = 0.324135

For x = 5,
P(X = 5) = 6C5 * 0.70^5 * 0.30^1
P(X = 5) = 0.302526

For x = 6,
P(X = 6) = 6C6 * 0.70^6 * 0.30^0
P(X = 6) = 0.117649

P(X ≥ 4) = 0.324135 + 0.302526 + 0.117649
P(X ≥ 4) = 0.744310

Hence, the probability that out of a group of 6 candidates at least 4 passed the examination is 0.744310 or 74.43%.
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The overall percentage of failure in a certain examination is 0.30. What is the probability that out of a group of 6 candidates at least 4 passed the examination?
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