Answer:
No, the events "brown hair" and "brown eyes" are not independent.
Step-by-step explanation:
The table that represents the hair and eye colors of thirty students of the fifth grade are given in table as:
Brown hair Blonde hair Total
Green eyes 9 6 15
Brown eyes 10 5 15
Total 19 11 30
No, the events "brown hair" and "brown eyes" are not independent.
Since, two events A and B are said to be independent if:
P(A∩B)=P(A)×P(B)
where P denotes the probability of an event.
Here we have:
A= students having brown hair.
B= students having brown eyes.
A∩B= students having both brown hair and brown eyes.
Now,
P(A)=19/30 (ratio of addition of first column to the total entries)
P(B)=15/30 ( ratio of addition of second row to the total entries)
Also,
P(A∩B)=10/30
Now as:
P(A∩B) ≠ P(A)×P(B)
Hence, the two events are not independent.
The scale factor is 5/3
9(5/3) = 15
Split up the interval [2, 5] into

equally spaced subintervals, then consider the value of

at the right endpoint of each subinterval.
The length of the interval is

, so the length of each subinterval would be

. This means the first rectangle's height would be taken to be

when

, so that the height is

, and its base would have length

. So the area under

over the first subinterval is

.
Continuing in this fashion, the area under

over the

th subinterval is approximated by

, and so the Riemann approximation to the definite integral is

and its value is given exactly by taking

. So the answer is D (and the value of the integral is exactly 39).