The probability of a successful optical alignment in the assembly of an optical data storage product is p = 0.6. assume the trials are independent. round your answers to four decimal places (e. g. 98.7654). (a) what is the probability that the 1st successful alignment requires exactly 4 trials? (b) what is the probability that the 1st successful alignment requires at most 4 trials? (c) what is the probability that the 1st successful alignment requires at least 4 trials?

a)

b)

c)

Step-by-step explanation:

The geometric distribution represents "the number of failures before you get a success in a series of Bernoulli trials. This discrete probability distribution is represented by the probability density function:"

Let X the random variable that measures the number os trials until the first success, we know that X follows this distribution:

Part a

For this case we want this probability

Part b

For this case we want this probability:

If we find the individual probabilities we got:

And replacing we have:

Part c

For this case at least 4 trials means that the random variable X needs to be 4 or more

And we found already the probabilities P(X=1),P(X=2) and P(X=3) so we just need to replace:

logv3(x+2)=1

rewrite in exponential form:

logb(x) = y is equal to b^y =x

b = 3, y = 1

now solve for x:

3^1 = x+2)

x = 3-2

x = 1

see the picture for the graph.

2.2°  

step-by-step explanation:

let θ = the angle of elevation.

tanθ =77/2020

tanθ = 0.038 12

      θ = tan⁻¹0.038 12

      θ = 2.2°