A company manufactures calculators in batches of 64 and there is a 4% rate of defects. Find the probability of getting exactly three defects in a batch.
Well, let's do the math. A batch contains 64 calculators and there is a 4% rate of defects. So, the probability of a calculator being defective is 4% or 0.04. The probability of a calculator being good is the complement of that, which is 1 - 0.04 = 0.96.
To find the probability of getting exactly three defects in a batch, we need to use the binomial distribution. The formula is:
P(X=k) = (n choose k) * p^k * (1-p)^(n-k)
Where:
- P(X=k) is the probability of getting exactly k defects,
- n is the number of trials or calculators in this case,
- p is the probability of getting a defect,
- k is the number of desired defects.
Using this formula, we can plug in the values:
P(X=3) = (64 choose 3) * (0.04)^3 * (0.96)^(64-3)
Now, you don't expect a clown to do all the calculations, do you? So, put on your thinking cap and get ready to crunch those numbers! It's time for some mathematical circus acrobatics!
To find the probability of getting exactly three defects in a batch, we can use the binomial probability formula:
P(X=k) = (n choose k) * p^k * (1-p)^(n-k)
Where:
P(X=k) is the probability of getting exactly k defects,
n is the total number of trials,
k is the number of successful trials (defects),
p is the probability of a single successful trial (defect).
In this case,
n = 64 (number of calculators in a batch)
k = 3 (number of defects)
p = 0.04 (probability of a defect)
Using the formula, we can calculate the probability:
P(X=3) = (64 choose 3) * 0.04^3 * (1-0.04)^(64-3)
First, let's calculate (64 choose 3):
(64 choose 3) = 64! / (3!(64-3)!)
= 64! / (3! * 61!)
(61! cancels out from the numerator and denominator)
= (64 * 63 * 62) / (3 * 2 * 1)
= 416,64
Now, let's substitute the values into the formula:
P(X=3) = (416,64) * 0.04^3 * (1-0.04)^(64-3)
≈ 0.3299
Therefore, the probability of getting exactly three defects in a batch is approximately 0.3299 or 32.99%.
To find the probability of getting exactly three defects in a batch, we can use the binomial probability formula.
The binomial probability formula is given by:
P(x) = (nCx) * (p^x) * (q^(n-x))
Where:
- P(x) is the probability of getting exactly x defects
- n is the number of trials (or calculators in this case) in a batch (n = 64)
- x is the number of successful outcomes (or defects in this case) in those trials (x = 3)
- p is the probability of success (or the defect rate) on a single trial (p = 4% = 0.04)
- q is the probability of failure on a single trial (q = 1 - p)
Now, let's plug in the values into the formula:
P(3) = (64C3) * (0.04^3) * (0.96^(64-3))
To calculate (64C3) or "64 choose 3," we need to use the combination formula which is given by:
(64C3) = 64! / (3! * (64-3)!)
Now, let's calculate (64C3) and plug in the values:
(64C3) = 64! / (3! * 61!)
= (64 * 63 * 62) / (3 * 2 * 1)
= 41664
Now we can calculate the probability:
P(3) = (41664) * (0.04^3) * (0.96^(64-3))
= (41664) * (0.0016) * (0.96^61)
≈ 0.2670
Therefore, the probability of getting exactly three defects in a batch is approximately 0.2670, or 26.70%.
The probability of seeing the first 61 in a batch good and the last 3 defective is (.96)^61 * (.04)^3 = X
Now then, count the number of ways a batch of 64 could have exactly 3 defectives. The formula for n-choose-c or 64-choose-3 is (n!/c!(n-c)!) (where ! means factorial).
64!/3!*61! = 41664.
Finally 41664*X = .221 or 22.1%
QED