Answer in fraction form = 6/11
Approximate answer in decimal form = 0.54545
Approximate answer in percent form = 54.545%
================================================================================
Explanation:
The uniform distribution is graphed as a rectangle. In this problem, the base of the rectangle goes from 12 to 27, which is a distance of 27-12 = 15 units. The height must be 1/15 to ensure that the area of the rectangle is 1.
area = base*height
area = 15*(1/15)
area = 1
The unit 1 refers to 100% probability
----------------
We are told that "given that the stock is greater than $16." which means we only focus on the portion of the rectangle from 16 to 27. The base is now 27-16 = 11 while the height remains the same
area = base*height
area = 11*(1/15)
area = 11/15
If we were to throw a random dart at the entire distribution, then the probability of landing on a stock price larger than $16 is 11/15
Let A = 11/15, which we'll use later.
---------------
Now we ask to find "the probability that the stock is less than $22". Keep in mind that we know the stock is greater than $16, so we consider the range from 16 to 22 which has a base of 22-16 = 6 and a height of 1/15, so
area = base*height
area = 6*(1/15)
area = 6/15
Let B = 6/15
---------------
Now divide B over A to get the conditional probability we're after
B/A = (6/15) divided by (11/15)
B/A = (6/15) * (15/11)
B/A = (6*15)/(15*11)
B/A = 6/11
B/A = 0.54545 approximately
--------------
Here's a visual. We know the stock is above $16, which is represented by this shaded red portion below
This red rectangle has base 11 and height 1/15, so its area is 11/15
If we want to find the probability of throwing a random dart in the purple region, then we would divide the purple rectangle area over the red rectangle area (from the above figure)
This purple rectangle has base 6 and height 1/15, so its area is 6/15
In other words,
answer = (area of purple rectangle)/(area of red rectangle)
answer = (area of rectangle BCFG)/(area of BDEG)
answer = (6/15)/(11/15)
answer = 6/11
answer = 0.54545 approximately
-----------------------------
Another approach:
Note how the rectangle ADEH is divided into 15*4 = 60 smaller rectangles because of the dashed grid lines
6*4 = 24 smaller rectangles are painted purple
11*4 = 44 smaller rectangles are painted red
24/44 = 6/11 represents the probability of throwing a random dart in the purple region given we know the dart landed in the red region.
-----------------------------
Yet another approach:
Divide the length of BC over BD
answer = (length of BC)/(length of BD)
answer = 6/11
This is the fastest method, but it does not use the rectangular uniform distribution. So it will depend on what your teacher wants you to show.