60% of candidates who took an examination passed .if 10 of the candidates are selected at random find correct to 3 significant figures the probability that exactly 4 passed
To find the probability that exactly 4 candidates passed, we can use the binomial probability formula. The formula is:
P(X = k) = (nCk) * p^k * (1-p)^(n-k)
Where:
- P(X = k) represents the probability of exactly k successes,
- n represents the number of trials or candidates selected,
- k represents the number of successes (in this case, the number of candidates who passed),
- p represents the probability of success (the probability that a candidate passes),
- (nCk) represents the number of possible combinations of selecting k successes out of n trials.
- (1-p) represents the probability of failure (the probability that a candidate fails).
Now let's calculate the probability:
n = 10 (since we are selecting 10 candidates)
k = 4 (we want exactly 4 candidates to pass)
p = 60% or 0.6 (which is the probability of passing)
To calculate the number of possible combinations (nCk), we use the formula:
nCk = n! / (k!(n-k)!)
Plugging in the values:
nCk = 10! / (4! * (10-4)!)
= 10! / (4! * 6!)
Calculating the factorial values:
10! = 10 * 9 * 8 * 7 * 6!
4! = 4 * 3 * 2 * 1
6! = 6 * 5 * 4 * 3 * 2 * 1
nCk = (10 * 9 * 8 * 7 * 6!) / ((4 * 3 * 2 * 1) * (6 * 5 * 4 * 3 * 2 * 1))
Now let's calculate the probability:
P(X = 4) = (nCk) * p^k * (1-p)^(n-k)
= [(10 * 9 * 8 * 7 * 6!) / ((4 * 3 * 2 * 1) * (6 * 5 * 4 * 3 * 2 * 1))] * (0.6^4) * ((1-0.6)^(10-4))
Simplifying the expression:
P(X = 4) = [(10 * 9 * 8 * 7) / (4 * 3 * 2 * 1)] * (0.6^4) * (0.4^6)
Calculating the probability:
P(X = 4) = [10 * 9 * 8 * 7] / [4 * 3 * 2 * 1] * 0.6^4 * 0.4^6
Using a calculator to evaluate the expression, the probability is approximately 0.29 (rounded to 3 significant figures). Therefore, the probability that exactly 4 candidates passed is 0.29.
If the events are independent, the probability of both/all events occurring is determined by multiplying the probabilities of the individual events.
(.6)^4 * (.4)^6 = ?