Sure! Let's go through each event step by step and identify the corresponding outcomes. Then, we'll determine the probabilities based on the given data.
### Event A: A tail on both the first and last tosses
We need outcomes where the first toss is T (tail) and the third toss is T (tail). Looking at the outcomes list:
- THT (Tail, Head, Tail)
- TTT (Tail, Tail, Tail)
There are 2 such outcomes out of 8 possible outcomes.
Probability of Event A:
Number of favorable outcomes = 2
Total number of outcomes = 8
[tex]\[ \text{Probability} = \frac{2}{8} = 0.25 \][/tex]
### Event B: A head on each of the last two tosses
We need outcomes where the second and third tosses are both H (head). Looking at the outcomes list:
- THH (Tail, Head, Head)
- HHH (Head, Head, Head)
- HHT (Head, Head, Tail)
There are 3 such outcomes out of 8 possible outcomes.
Probability of Event B:
Number of favorable outcomes = 3
Total number of outcomes = 8
[tex]\[ \text{Probability} = \frac{3}{8} = 0.375 \][/tex]
### Event C: Alternating tail and head (with either coming first)
We need outcomes where the tosses alternate between T (tail) and H (head):
- HTH (Head, Tail, Head)
- THT (Tail, Head, Tail)
There are 2 such outcomes out of 8 possible outcomes.
Probability of Event C:
Number of favorable outcomes = 2
Total number of outcomes = 8
[tex]\[ \text{Probability} = \frac{2}{8} = 0.25 \][/tex]
### Summary
Let's summarize these probabilities in the given table format:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|c|c|c|}
\hline \text{} & \multicolumn{8}{|c|}{ \text{Outcomes} } & \multirow{2}{*}{ \text{Probability} } \\
\hline \text{} & \text{HHH} & \text{HHT} & \text{HTH} & \text{HTT} & \text{THH} & \text{THT} & \text{TTH} & \text{TTT} & \\
\hline \begin{array}{l}
\text{Event A: A tail on both the first and the} \\
\text{last tosses}
\end{array} & & & & & & \checkmark & & \checkmark & 0.25 \\
\hline \begin{array}{l}
\text{Event B: A head on each of the last two} \\
\text{tosses}
\end{array} & \checkmark & \checkmark & & & \checkmark & & & & 0.375 \\
\hline \begin{array}{l}
\text{Event C: Alternating tail and head (with} \\
\text{either coming first)}
\end{array} & & & \checkmark & & & \checkmark & & & 0.25 \\
\hline
\end{array}
\][/tex]
Thus, the probabilities for Event A, Event B, and Event C are [tex]\(0.25\)[/tex], [tex]\(0.375\)[/tex], and [tex]\(0.25\)[/tex] respectively.
