To solve this problem, we need to calculate the probability function of the number of heads when a coin is tossed four times.

1. Identify the Sample Space: When a coin is tossed four times, each toss has two possible outcomes: Heads (H) or Tails (T). So, the sample space has $2^4 = 16$ possible outcomes.

2. Define the Random Variable $X$: Let $X$ be the number of heads obtained in four tosses.

3. Possible Values of $X$: The number of heads (X) can be 0, 1, 2, 3, or 4.

4. Calculate Probabilities for Each Value of $X$:

   • Use the binomial distribution formula: $P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}$ where $n = 4$ (number of tosses), $k$ is the number of heads, and $p = 0.5$ (probability of getting heads in each toss).

5. Fill in the Probability Table:

Let’s calculate each probability.

• For $X = 0$: $P(X = 0) = \binom{4}{0} (0.5)^0 (0.5)^4 = 1 \times (0.5)^4 = 0.0625$
• For $X = 1$: $P(X = 1) = \binom{4}{1} (0.5)^1 (0.5)^3 = 4 \times 0.5 \times 0.125 = 0.25$
• For $X = 2$: $P(X = 2) = \binom{4}{2} (0.5)^2 (0.5)^2 = 6 \times 0.25 \times 0.25 = 0.375$
• For $X = 3$: $P(X = 3) = \binom{4}{3} (0.5)^3 (0.5)^1 = 4 \times 0.125 \times 0.5 = 0.25$
• For $X = 4$: $P(X = 4) = \binom{4}{4} (0.5)^4 = 1 \times 0.0625 = 0.0625$

6. Tabular Form of Probability Function:

Would you like more details on how each probability was calculated, or do you have any questions?

1. What is the binomial distribution, and why is it used here?
2. How do you calculate binomial probabilities for different numbers of trials?
3. What is the probability of getting exactly three heads in four coin tosses?
4. How does changing the probability of heads affect the distribution?
5. How would this table change if the coin was tossed five times?

Tip: Remember that binomial probabilities rely on the concept of independent events, where each coin toss does not affect the next.