Answer:
(a) P(X = 4) = 0.0189
(b) P(X 4) = 0.9919
(c) P(X 4) = 0.027
Step-by-step explanation:
We are given that the probability of a successful optical alignment in the assembly of an optical data storage product is p = 0.7 .
Since we have to find the probabilities for 1st successful alignments, so the probability distribution that we will use here is Geometric distribution.
<u>Geometric distribution</u> is used when we are interested in knowing the chances of our first success.
The probability distribution of geometric distribution is given by;
where, = number of trials
k = first success = 1
p = probability of getting success = 0.70
So, X ~ Geo(p = 0.7)
(a) Probability that the 1st successful alignment requires exactly 4 trials is given by = P(X = 4)
Here, = 4, p = 0.7 and k = 1
So, P(X = 4) = =
= 0.0189
(b) Probability that the 1st successful alignment requires at most 4 trials is given = P(X 4)
P(X 4) = P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)
=
=
= 0.7 + 0.21 + 0.063 + 0.0189 = 0.9919
(c) Probability that the 1st successful alignment requires at least 4 trials is given by = P(X 4)
P(X 4) = 1 - P(X < 4) = 1 - P(X
3)
P(X 4) = 1 - P(X = 1) - P(X = 2) - P(X = 3)
=
=
= 1 - 0.7 - 0.21 - 0.063 = 0.027