Random Theory
More examples
Example 1- Circuit operation
• The following circuit operates only if there is a path of functional
devices from left to right. The probability that each device functions is
shown on the graph. Assume that devices fail independently. What is
the probability that the circuit operates?
Answer
• P(A) = 0.8 and P(B) = 0.9, so P(A and B) = P(A).P(B) = 0.8x 0.9 = 0.72
Example 2
Example 2 – Continue
Example 3
• An optical inspection system is to distinguish among different part
types. The probability of a correct classification of any part is 0.98.
Suppose that five parts are inspected and that the classifications are
independent. Let the random variable X denote the number of parts
that are correctly classified.
(1) What is the probability that all parts are correctly classified?
(2) What is the probability that exactly one part is incorrectly classified?
Example 4
• Each sample of water has a 10% chance of containing a particular
organic pollutant. Assume that the samples are independent with
regard to the presence of the pollutant. Find the probability that in
the next 18 samples, exactly 2 contain the pollutant.
Answer
• Let X the number of samples that contain the pollutant in the next 18
samples analyzed. Then X is a binomial random variable with p = 0.1
and n = 18.
Example 5
Answer
• (a) 𝑃 𝑋 ≤ 2 = 1
• (b) 𝑃 𝑋 > −2 =
2
8
• (c) 𝑃 −1 ≤ 𝑋 ≤ 1
2 2 1 7
+ + + =
8 8 8 8
2 2 2 3
= + + =
8 8 8 4
Example 6
• The number of customers arriving at a grocery store can be modeled
by a Poisson process with intensity λ = 10 customers per hour.
• a) Find the probability that there are two customers between 10 and
10:20
• b) What is the probability that at least 30 minutes will pass until the
next customers arrives?
Answer
• a)
• b) As we are talking about the time, we are now following the
∞
1
exponential distribution 𝑃 𝑋 ≥ = ‫׬‬1/2 10𝑒 −10𝑥 𝑑𝑥 =e-10(1/2)
2
Example 7
• Let X to be continues random variable having range [0, 2] and density
. Find E(X)
Answer
Example 8
• The diameter of a shaft in an optical storage drive is normally
distributed with mean 0.2508 inch and standard deviation 0.0005
inch. The specifications on the shaft are 0.2500 +/- 0.0015 inch. What
proportion of shafts conforms to specifications?
Answer
• Let X denote the shaft diameter in inches. The requested probability
is shown as follows: