If 5% of electric bulb manufactured by a company are defective find the probability that in a sample of 100 bulbs.
(1)None if defective
(2) 5 bulbs are defective
e -5=0.007
To find the probability in each case, we can use the concept of binomial probability.
The binomial probability formula is:
P(x) = nCx * p^x * q^(n-x)
Where:
- P(x) is the probability of getting x successes in n trials
- nCx is the number of combinations of n items taken x at a time
- p is the probability of success in one trial
- q is the probability of failure in one trial (1-p)
- x is the number of successes
Let's solve each case separately using the formula:
(1) None are defective:
In this case, we want to find the probability that none out of 100 bulbs are defective.
So x = 0 (no successes), n = 100 (number of trials), p = 0.05 (probability of success), and q = 0.95 (probability of failure).
Using the formula:
P(x=0) = 100C0 * 0.05^0 * 0.95^(100-0)
100C0 = 1 (since choosing 0 out of 100 bulbs can be done in only one way)
P(x=0) = 1 * 1 * 0.95^100 ≈ 0.006 ≈ 0.6%
Therefore, the probability that none out of 100 bulbs are defective is approximately 0.6%.
(2) Five bulbs are defective:
In this case, we want to find the probability that exactly 5 out of 100 bulbs are defective.
So x = 5, n = 100, p = 0.05, and q = 0.95.
Using the formula:
P(x=5) = 100C5 * 0.05^5 * 0.95^(100-5)
100C5 = (100!)/(5!(100-5)!) = 75287520
P(x=5) = 75287520 * 0.05^5 * 0.95^95 ≈ 0.036
Therefore, the probability of exactly 5 out of 100 bulbs being defective is approximately 0.036.