If the probability that the average freshman will not complete a 5 year engineering course is 2/5, what is the probability that 4 freshmen at least 3 will complete the 5 year course.
prob(exactly 3 of 5 will complete)
= C(5,3) (2/5)^3 (3/5)^2
= 10(72/3125)
= 144/625 or .2304
prob(exactly 4 of 5 will complete)
= ....
prob(exactly 5 of 5 will complete)
= ...
add them up
Well, let's break it down step by step. The probability that a single freshman will NOT complete the 5-year engineering course is 2/5. So, the probability that a single freshman WILL complete the course is 1 - 2/5 = 3/5.
Now, let's look at the situation where we have 4 freshmen. We want to find the probability that AT LEAST 3 of them will complete the course.
To calculate this, we can consider the possible combinations: 3 students completing and 1 not completing, or 4 students completing.
For 3 students completing and 1 not completing, we need to calculate the probability for each scenario and sum them up:
(3/5) * (3/5) * (3/5) * (2/5) = 54/625
For 4 students completing, we simply need to calculate the probability:
(3/5) * (3/5) * (3/5) * (3/5) = 81/625
Now, let's sum up the probabilities:
54/625 + 81/625 = 135/625
Therefore, the probability that at least 3 out of 4 freshmen will complete the 5-year engineering course is 135/625.
To find the probability that at least 3 out of 4 freshmen will complete the 5-year engineering course, we can use the binomial probability formula.
The formula for the probability of exactly k successes in n independent Bernoulli trials, each with probability of success p, is given by:
P(X = k) = (n C k) * p^k * (1 - p)^(n - k)
Where:
n = number of trials
k = number of successful trials
p = probability of success in each trial
(1 - p) = probability of failure in each trial
In this case, we want to find the probability of at least 3 out of 4 freshmen completing the course. This means we need to find the probability of exactly 3 students completing the course, plus the probability of exactly 4 students completing the course.
Let's calculate the probabilities step-by-step:
Probability of exactly 3 completing:
P(X = 3) = (4 C 3) * (2/5)^3 * (3/5)^(4-3)
(4 C 3) = 4! / (3! * (4-3)!) = 4
P(X = 3) = 4 * (2/5)^3 * (3/5)^1
P(X = 3) = 4 * 8/125 * 3/5
P(X = 3) = 96/625
Probability of exactly 4 completing:
P(X = 4) = (4 C 4) * (2/5)^4 * (3/5)^(4-4)
(4 C 4) = 4! / (4! * (4-4)!) = 1
P(X = 4) = 1 * (2/5)^4 * (3/5)^0
P(X = 4) = 1 * 16/625 * 1
P(X = 4) = 16/625
Now, we can sum up the probabilities of exactly 3 and exactly 4 completing to find the probability of at least 3 out of 4 completing the course:
P(at least 3) = P(X = 3) + P(X = 4)
P(at least 3) = 96/625 + 16/625
P(at least 3) = 112/625
Therefore, the probability that at least 3 out of 4 freshmen will complete the 5-year engineering course is 112/625.
To find the probability that at least 3 out of 4 freshmen will complete the 5-year engineering course, we can use the binomial probability formula. Let's break down the solution step by step:
Step 1: Determine the probability that a single freshman will not complete the 5-year course.
Given that the probability that a freshman will not complete the course is 2/5, the probability that a freshman will complete the course (P) can be found by subtracting the probability of not completing (2/5) from 1:
P = 1 - 2/5
P = 3/5
Step 2: Apply the binomial probability formula.
The binomial probability formula is given by:
P(x) = (nCx) * (p^x) * (q^(n-x))
where:
- P(x) represents the probability of "x" successes
- nCx represents the number of ways to choose "x" successes from "n" trials (given by the combination formula, nCx = n! / (x! * (n-x)!)
- p represents the probability of a single success
- q represents the probability of a single failure (1 - p)
- x represents the number of successes
- n represents the total number of trials
In this case, we want to find the probability that at least 3 out of 4 freshmen will complete the course. So, we need to calculate the probability of 3 successes and the probability of 4 successes, and then sum those probabilities.
Step 3: Calculate the probability of 3 successes.
P(3) = (4C3) * ((3/5)^3) * ((2/5)^(4-3))
P(3) = 4 * (3/5)^3 * (2/5)^1
P(3) ≈ 0.276
Step 4: Calculate the probability of 4 successes.
P(4) = (4C4) * ((3/5)^4) * ((2/5)^(4-4))
P(4) = 1 * (3/5)^4 * (2/5)^0
P(4) ≈ 0.1296
Step 5: Calculate the probability of at least 3 successes.
P(at least 3) = P(3) + P(4)
P(at least 3) ≈ 0.276 + 0.1296
P(at least 3) ≈ 0.4056
Therefore, the probability that at least 3 out of 4 freshmen will complete the 5-year engineering course is approximately 0.4056.