Given Problem:

You are tossing a fair coin 3 times, and you want to compute the probability distribution for the number of heads (denoted by X) that appear. Additionally, we will calculate the expected number of heads and determine the expected profit from the game.

---

Step 1: Sample Space

The sample space for the three coin tosses can be written as:

• TTT (0 heads)
• TTH (1 head)
• THT (1 head)
• HTT (1 head)
• THH (2 heads)
• HTH (2 heads)
• HHT (2 heads)
• HHH (3 heads)

Step 2: Probability of Each Event

Each toss of the coin is independent and has a 50% chance of landing heads (H) and a 50% chance of landing tails (T). So, the probability of each specific sequence in the sample space is:

$P(\text{each sequence}) = \frac{1}{8}$

Step 3: Distribution of X (Number of Heads)

For each outcome, the number of heads (X) is counted. Let's list the probability distribution for X:

• X = 0 heads (TTT): 1 event
• X = 1 head (TTH, THT, HTT): 3 events
• X = 2 heads (THH, HTH, HHT): 3 events
• X = 3 heads (HHH): 1 event

Now, we can compute the probability of each value of X by dividing the number of favorable outcomes by the total number of possible outcomes (which is 8):

$P(X = 0) = \frac{1}{8}, \quad P(X = 1) = \frac{3}{8}, \quad P(X = 2) = \frac{3}{8}, \quad P(X = 3) = \frac{1}{8}$

Thus, the probability distribution for X is:

x	0	1	2	3
P(X = x)	1/8	3/8	3/8	1/8

---

Step 4: Expected Number of Heads (a)

The expected value (or mean) of a discrete random variable is calculated by:

$E(X) = \sum_{x} x \cdot P(X = x)$

Using the probability distribution:

$E(X) = 0 \cdot \frac{1}{8} + 1 \cdot \frac{3}{8} + 2 \cdot \frac{3}{8} + 3 \cdot \frac{1}{8}$

$E(X) = 0 + \frac{3}{8} + \frac{6}{8} + \frac{3}{8} = \frac{12}{8} = 1.5$

Thus, the expected number of heads is 1.5.

---

Step 5: Expected Profit (b)

Now, let's calculate the expected profit from playing the game.

1. Initial cost to play: $10
2. Payment per head: $6

The total payment you receive depends on the number of heads that appear. So, the expected total payment is:

$\text{Expected payment} = 6 \cdot E(X) = 6 \cdot 1.5 = 9$

Now, to find the expected profit:

$\text{Expected profit} = \text{Expected payment} - \text{Initial cost}$

$\text{Expected profit} = 9 - 10 = -1$

Thus, the expected profit from this game is -$1.

---

Final Answer:

• a) The expected number of heads that will appear is 1.5.
• b) The expected profit from playing the game is -$1.

---

Do you need further explanation or clarification on any part of the solution?

---

Related Questions:

1. How would the expected profit change if the payment per head was $5 instead of $6?
2. If you played the game multiple times, what is the total expected profit after 10 games?
3. How would the probability distribution change if the coin was biased?
4. How do you calculate the variance and standard deviation for this type of probability distribution?
5. How can we use this type of game theory to determine optimal strategies for decision-making?

Tip: To understand expected values better, you can think of them as the "long-term average" result of many repetitions of the game.