The probability that a machine produces a defec tive item is

The probability that a machine produces a defec- tive item is 0.01. Each item is checked as it is produced. Assume that these are independent trials, and compute the probability that at least 100 items must be checked to find one that is defective.

Solution

AS WE ARE LOOKING FOR 1ST SUCCESS THAT IS 1ST DEFECTIVE AMONG 100 WE CAN USE GEOMETRIC DISTRIBUTION.

Let X be the number of trials until the first success.The Geometric is looking for the number of trials before the first success.

X has the Geometric Distribution with success probability p then:

X ~ Geometric(p)

P(X = x) = p * (1 - p) ^ (x - 1) for x = 1, 2, 3, 4, ....
P(X = x) = 0 otherwise.

the probability mass function is derived by looking at having x - 1 failures and then 1 success.


The Variance of the Geometric is (1 - p) / p^2

X ~ Geometric( 0.01 )
E(X) = 100
Var(X) = 9900

P( X 100)
= 1 - P(X < 100)

=

. . . 99
1 - P(X = t)
. . . t = 1

= 1 - 0.01 * 0.99 ^ (t - 1)

= 1 - 0.6302704

= 0.3697296