The probability that a random student at elmville college is a freshman is 0.3; a sophomore, 0.25; and a junior or senior, 0.45. the probability that a freshman will major in engineering is 0.15; a sophomore, 0.2; and a junior or senior, 0.3. what is the probability that a random student at this college majors in engineering?

Probability of an event is the measure of its chance of occurrence. The probability that a random student at this college majors in engineering is 0.24

What is chain rule in probability?

For two events A and B, by chain rule, we have:

What is law of total probability?

Suppose that the sample space is divided in n mutual exclusive and exhaustive events tagged as

Then, suppose there is event A in sample space.

Then probability of A's occurrence can be given as

Using the chain rule, we get

For the given case, let we consider events:

A = Student randomly selected is freshmanB =  Student randomly selected is sophomoreC =  Student randomly selected is junior or seniorD = Student randomly selected majors in engineering.

Then, we have:

P(A)  = 0.3P(B) = 0.25P(C) = 0.45P(D|A) = 0.15 (Since A had happened already(as they said student is known to be a freshmen) and then we were calculating D's probability. Its notation is P(D|A) )P(D|B) = 0.2P(D|C) = 0.3

The needed probability is

P(random student at this college majors in engineering) = P(D)

Using the law of total probability, we get:

P(D) = P(A).P(D|A) + P(B).P(D|B) + P(C).P(D|C)

P(D) = 0.045 + 0.06 + 0.135 = 0.24

Thus,

The probability that a random student at this college majors in engineering is 0.24

Learn more about law of total probability here:

link

Let denote the event that a student is a freshman, a sophomore, a junior, and a senior.

Let denote the event that a student majors in engineering. By the law of total probability,



By the definition of conditional probability, we can expand each of these intersection probabilities to get

L:H=3:2

L=1.8
1.8=3 units
divide by 3
0.6=1unit

2unit=height
0.6 times 2=height=1.2

height=1.2yards