Suppose that you

flip a fair coin (P(H)=P(T)=1
2 ) three times and you record if it landed on heads, H, or tails,
T.
(a) What is the sample space of this experiment? What is the probability of each event?
(b) [1 pt] Let X be the number of times that you observe heads. What type of random variable is X?
(c) What is the probability distribution of X?

By the way, it is not clear to me from the wording of the question if h, t, t is the same as t, h , t or not

the first way there are 6 possible outcomes. The second way there are 8

If order does not matter
1 h h h
2 h h t
3 h t t
4 t t t

probability of each event here is 1/4

if order does matter

1 h h h
2 h h t
3 h t h
4 h t t
5 t t t
6 t t h
7 t h t
8 t h h
probability of each event here is 1/8

In the second list, listing all possible outcomes there is one way to get three heads p(3h) = 1/8

three ways of getting two heads
p(2h) = 3/8
three ways of getting 1 head
p(1h) = 3/8
one way of getting 0 heads
p (0h) = 1/8

let's look at a binomial distribution where p(h) = 1/2
n = number of trials = 3
k = number of successes (number of heads)
p(h = k) = C(n,k) p^k (1-p)^k

p(h=3) = c(3,3)(1/2)^3(1/2)^0
= 3!/[3!*0!](1/8)(1) = 1/8

p(h=1) = C(3,1)(1/2)(1/2)^2
c(3,1) = 3!/[1!(3-1)!] = 3*2/2 = 3
so
p(h=1) = 3(1/2)(1/4) = 3/8

you will find that p(h=2) = C(3,2)((1/2)^2(1/2)^1 = 3/8
etc

(a) The sample space of this experiment is the set of all possible outcomes when flipping a coin three times. Since each coin flip can result in either heads (H) or tails (T), there are 2 possible outcomes for each flip. Therefore, the sample space consists of all possible combinations of H and T for three flips:

Sample space: {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}

The probability of each event in the sample space can be determined by considering the probability of heads (H) and tails (T) for each individual coin flip. Since the coin is fair, the probability of getting heads or tails is 1/2 (or 0.5) for each flip. Therefore, each event in the sample space has a probability of (1/2)^3 = 1/8 (or 0.125).

(b) The random variable X, defined as the number of times heads is observed, is a discrete random variable. It represents the count of a specific outcome (in this case, heads) in a set number of trials (in this case, three coin flips).

(c) To determine the probability distribution of X, we need to find the probability of each possible value of X (0, 1, 2, and 3). We can do this by calculating the probability of each combination of heads (H) and tails (T) in the sample space and summing the probabilities for each value of X.

- X = 0: The probability of observing 0 heads is the same as the probability of getting all tails in three flips. There is only one event in the sample space where this occurs: TTT. So, P(X=0) = 1/8.

- X = 1: The probability of observing 1 head is the sum of the probabilities of all events with exactly one heads. There are three such events: HTT, THT, and TTH. So, P(X=1) = 3/8.

- X = 2: The probability of observing 2 heads is the sum of the probabilities of all events with exactly two heads. There are three such events: HHT, HTH, and THH. So, P(X=2) = 3/8.

- X = 3: The probability of observing 3 heads is the same as the probability of getting all heads in three flips. There is only one event in the sample space where this occurs: HHH. So, P(X=3) = 1/8.

Therefore, the probability distribution of X is:
P(X=0) = 1/8
P(X=1) = 3/8
P(X=2) = 3/8
P(X=3) = 1/8