A factory finds out that on the average 20% of bolts produced by the given machine will be defective.if 10 bolts are selected at random from the day's production of this machine,find the probability that 1.exactly 2 will be defective.
(this question is base on probability and combinatorial Analysis)
p good = .2
p not good = .8
this is a binomial distribution problem
P(n,k) = C(n,k) p^k (1-p)^(n-k)
where
C(n,k) = n!/[k!(n-k)!]
here for example
P(10,1) = C(10,1).2^1 .8^9
C(10,1)= 10!/[1(9)!] = 10
so
P(10,1) = 10 * .2 * .8^9
= .268
similarly
C(10,2) = 10!/[2(8!] = 10*9/2 =45
so
P(10,2) = 45 .2^2 .8^8
= .302
To find the probability that exactly 2 out of 10 bolts will be defective, we can use the binomial probability formula:
P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)
Where:
P(X = k) is the probability of exactly k successes.
C(n, k) is the combination formula for selecting k items from a set of n items.
p is the probability of success in a single trial.
n is the total number of trials.
In this case:
k = 2 (exactly 2 defective bolts)
n = 10 (total number of bolts selected)
p = 0.2 (probability of a bolt being defective)
Let's calculate it step-by-step:
Step 1: Calculate the combination C(n, k)
C(10, 2) = 10! / ((10 - 2)! * 2!) = 10! / (8! * 2!) = (10 * 9) / (2 * 1) = 45
Step 2: Calculate the probability of having exactly 2 defective bolts
P(X = 2) = C(10, 2) * (0.2)^2 * (1 - 0.2)^(10 - 2)
= 45 * (0.2)^2 * (0.8)^8
= 45 * 0.04 * 0.16777216
= 0.302052
Therefore, the probability that exactly 2 bolts out of the 10 selected will be defective is approximately 0.302, or 30.2%.
To find the probability that exactly 2 out of 10 bolts will be defective, we can use the binomial probability formula. The formula is given by:
P(x) = C(n, x) * p^x * (1 - p)^(n - x)
Where:
P(x) is the probability of exactly x successes
C(n, x) is the number of ways to choose x successes from n trials
p is the probability of success in a single trial
n is the number of trials
In this case, we have:
p = 0.20 (probability of a bolt being defective)
n = 10 (number of bolts selected)
Now we can plug these values into the formula and calculate the probability. Let's do it step by step.
Step 1: Calculate C(n, x)
C(n, x) represents the number of ways to choose x successes from n trials. The formula for this is:
C(n, x) = (n!)/[(x!) * (n - x)!]
Here, "!" denotes factorial.
In our case, n = 10 and x = 2:
C(10, 2) = (10!)/[(2!) * (10 - 2)!]
= (10!)/[(2!) * 8!]
= (10 * 9)/2
= 45
Step 2: Plug in the values into the binomial probability formula
P(2) = C(10, 2) * p^2 * (1 - p)^(10 - 2)
= 45 * (0.20)^2 * (0.80)^8
Step 3: Calculate the result
P(2) = 45 * 0.04 * 0.1677
= 0.3015
Therefore, the probability that exactly 2 out of the 10 bolts will be defective is approximately 0.3015 or 30.15%.