If 5% of the bolts made by an automotive factory are defective , what is the probability that in a shipment of 200 bolts, there are 15 defective bolts?
Use the binomial probability formula
n=200 and p = 0.05
a. 0.081
b. 0.034
c. 0.313
d. 0.054
I read that as exactly 15 of the 200 to be defective
= C(200,15)( .05)^15 (.95)^185
= .0337.. or their answer of .034
Wow, those were some very large and very small numbers,
good thing my calculator has more than one memory location.
Yeah, is 15 defective, sorry for the question mark
Thank You!
200C15 .95^185 .05^15
To find the probability that in a shipment of 200 bolts, there are 15 defective bolts, we can use the binomial probability formula.
The binomial probability formula is:
P(X = k) = (n C k) * p^k * (1 - p)^(n - k)
Where:
P(X = k) is the probability of getting exactly k successes.
n is the total number of trials or observations.
k is the number of successes we want to find the probability for.
p is the probability of success in a single trial.
(1 - p) is the probability of failure in a single trial.
(n C k) is the binomial coefficient, which represents the number of ways to choose k successes out of n trials.
In this case, n = 200 (total number of bolts in the shipment) and p = 0.05 (probability that a bolt is defective). We want to find P(X = 15), so k = 15.
Now let's calculate the probability using the formula:
P(X = 15) = (200 C 15) * 0.05^15 * (1 - 0.05)^(200 - 15)
To calculate the binomial coefficient (n C k), we use the following formula:
(n C k) = n! / (k! * (n - k)!)
Applying this formula:
(200 C 15) = 200! / (15! * (200 - 15)!)
Calculating the factorials:
200! = 200 * 199 * 198 * ... * 187 * 186 * ... * 3 * 2 * 1
15! = 15 * 14 * 13 * ... * 3 * 2 * 1
(200 - 15)! = 185 * 184 * ... * 3 * 2 * 1
Now we can calculate:
(200 C 15) = (200 * 199 * 198 * ... * 187 * 186 * ... * 3 * 2 * 1) / ((15 * 14 * 13 * ... * 3 * 2 * 1) * (185 * 184 * ... * 3 * 2 * 1))
Once we have the value for (200 C 15), we can substitute it back into the formula:
P(X = 15) = ((200 C 15) * 0.05^15 * (1 - 0.05)^(200 - 15))
Calculating this expression will give us the final answer, which should be one of the options provided:
a. 0.081
b. 0.034
c. 0.313
d. 0.054