Let's solve the problem step by step to find the probability distribution for the number of heads occurring in three coin tosses.
### Step 1: Identify the sample space
The sample space, [tex]\( S \)[/tex], for tossing a coin three times consists of all possible sequences of heads (H) and tails (T). There are [tex]\( 2^3 = 8 \)[/tex] possible outcomes:
[tex]\[
S = \{ HHH, HHT, HTH, HTT, THH, THT, TTH, TTT \}
\][/tex]
### Step 2: Define the random variable
Let [tex]\( X \)[/tex] be the random variable representing the number of heads in three coin tosses. [tex]\( X \)[/tex] can take on the values 0, 1, 2, or 3.
### Step 3: Count the occurrences of each value of [tex]\( X \)[/tex]
We count how many outcomes in the sample space correspond to each possible number of heads.
- 0 heads:
- Outcome: TTT
- Number of outcomes: 1
- 1 head:
- Outcomes: HTT, THT, TTH
- Number of outcomes: 3
- 2 heads:
- Outcomes: HHT, HTH, THH
- Number of outcomes: 3
- 3 heads:
- Outcome: HHH
- Number of outcomes: 1
### Step 4: Calculate the probabilities
To find the probability of each value of [tex]\( X \)[/tex], we divide the number of outcomes with that value by the total number of outcomes in the sample space.
- Probability that [tex]\( X = 0 \)[/tex] (0 heads):
[tex]\[
P(X = 0) = \frac{\text{Number of outcomes with 0 heads}}{\text{Total number of outcomes}} = \frac{1}{8} = 0.125
\][/tex]
- Probability that [tex]\( X = 1 \)[/tex] (1 head):
[tex]\[
P(X = 1) = \frac{\text{Number of outcomes with 1 head}}{\text{Total number of outcomes}} = \frac{3}{8} = 0.375
\][/tex]
- Probability that [tex]\( X = 2 \)[/tex] (2 heads):
[tex]\[
P(X = 2) = \frac{\text{Number of outcomes with 2 heads}}{\text{Total number of outcomes}} = \frac{3}{8} = 0.375
\][/tex]
- Probability that [tex]\( X = 3 \)[/tex] (3 heads):
[tex]\[
P(X = 3) = \frac{\text{Number of outcomes with 3 heads}}{\text{Total number of outcomes}} = \frac{1}{8} = 0.125
\][/tex]
### Step 5: Summarize the probability distribution
We can now compile our results into a table showing the probability distribution for [tex]\( X \)[/tex]:
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
X & 0 & 1 & 2 & 3 \\
\hline
p(X) & 0.125 & 0.375 & 0.375 & 0.125 \\
\hline
\end{array}
\][/tex]
Thus, the probability distribution for the number of heads occurring in three coin tosses is:
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
X & 0 & 1 & 2 & 3 \\
\hline
p(X) & \frac{1}{8} & \frac{3}{8} & \frac{3}{8} & \frac{1}{8} \\
\hline
\end{array}
\][/tex]
