**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