PFA

Suppose a coin is flipped and a die is rolled. Let E1 denote the event "the coin shows a tail", let E2 denote the event "the dies shows a 3", and let E3 denote the event "the coin shows heads and the die shows an odd number". Are E1 and E2 mutually exclusive?

E1 and E2 are NOT mutually exclusive because both can happen at the same time. If they were mutually exclusive, then the coin showing tails would somehow prevent the die showing a three and vice versa. The coin and the die have no influence over each other and a tail and a 3 can happen simultaneously, so the events are not mutually exclusive.

A family of four children. Assume that it is equally probable for a boy or a girl to be born. What is the probability of exactly two girls?

There are eight possibilities for there to be 2 girls in a family of 4 children:

BBGG
BGBG
BGGB
GBBG
GBGB
GGBB

Each of these arrangements has a probability of (0.5)(0.5)(0.5)(0.5) = 0.0625 (each of the 0.5 is the probability that either a boy or a girl was born at that position in the birth order).

Since there are eight possibilities, each with a probability of 0.0625, the total probability for exactly two girls is:

(0.0625)(8) = 0.5

Find a recurrence relation and initial conditions that generate a sequence that begins with the given term.

3,6,9,15,24,39,...

Let's look what happens if you subtract each number from the next one:

6 - 3 = 3
9 - 6 = 3
15 - 9 = 6
24 - 15 = 9
39 - 24 = 15

It seems as if the next term in the sequence is made by adding the two previous ones (3 + 6 = 9, 6 + 9 = 15, etc.).
There was no way to generate 6, because there was only one number that came before it. So, if we designate the numbers in the sequence as a0, a1, a2, ...., an, ..., then the initial conditions are a0 = 3 and a1 = 6. All the other numbers can be generated using the relation:

an+1 = an + an-1