Sure, let's break down this problem step by step. We start by analyzing the given outcomes, then we determine the events, and finally, we calculate the probabilities for each event.
List of Outcomes:
1. OEE
2. OEO
3. EEO
4. OOO
5. EOE
6. EOO
7. EEE
8. OOE
Each outcome is equally likely, so each outcome has a probability of [tex]\(\frac{1}{8}\)[/tex].
Event A: An odd number on each of the last two rolls
We need to identify the outcomes where the second and third rolls are both odd.
- Outcomes: OOO, EOO
Total favorable outcomes: 2
Probability of Event A: [tex]\(\frac{2}{8} = 0.25\)[/tex]
Event B: Two or more even numbers
We need to identify the outcomes where there are at least two even numbers.
- Outcomes: OEE, OEO, EEO, EOE, EEE
Total favorable outcomes: 4
Probability of Event B: [tex]\(\frac{4}{8} = 0.5\)[/tex]
Event C: An even number on the last roll or the second roll (or both)
We need to identify the outcomes where either the second roll, the third roll, or both are even.
- Outcomes: OEE, OEO, EEO, EOE, EEE, OOE
Total favorable outcomes: 6
Probability of Event C: [tex]\(\frac{6}{8} = 0.75\)[/tex]
So, our table should now be filled out like so:
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline & \multicolumn{8}{|c|}{ Outcomes } & \multirow{2}{*}{ Probability } \\
\hline & OEE & OEO & EEO & OOO & EOE & EOO & EEE & [tex]$OOE$[/tex] & \\
\hline \begin{tabular}{c}
Event A: An odd number on each \\
of the last two rolls
\end{tabular} & & & & X & & X & & & 0.25\\
\hline Event B: Two or more even numbers & X & X & X & & X & & X & & 0.5 \\
\hline \begin{tabular}{c}
Event C: An even number on the last \\
roll or the second roll (or both)
\end{tabular} & X & X & X & & X & & X & X & 0.75 \\
\hline
\end{tabular}
