A number cube is rolled three times. An outcome is represented by a string of the sort OEE (meaning an odd number on the first roll, an even number on the second roll, and an even number on the third roll). The 8 outcomes are listed in the table below. Note that each outcome has the same probability.

For each of the three events in the table, check the outcome(s) that are contained in the event. Then, in the last column, enter the probability of the event.

\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline
& \multicolumn{8}{|c|}{Outcomes} & \multirow{2}{*}{Probability} \\
\cline{2-9}
& OEE & OEO & EEO & OOO & EOE & EOO & EEE & OOE & \\
\hline
Event A: An odd number on each of the last two rolls & [tex]$\square$[/tex] & [tex]$\square$[/tex] & [tex]$\square$[/tex] & [tex]$\square$[/tex] & [tex]$\square$[/tex] & [tex]$\square$[/tex] & [tex]$\square$[/tex] & [tex]$\square$[/tex] & \\
\hline
Event B: Two or more even numbers & [tex]$\square$[/tex] & [tex]$\square$[/tex] & [tex]$\square$[/tex] & [tex]$\square$[/tex] & [tex]$\square$[/tex] & [tex]$\square$[/tex] & [tex]$\square$[/tex] & [tex]$\square$[/tex] & \\
\hline
Event C: An even number on the last roll or the second roll (or both) & [tex]$\square$[/tex] & [tex]$\square$[/tex] & [tex]$\square$[/tex] & [tex]$\square$[/tex] & [tex]$\square$[/tex] & [tex]$\square$[/tex] & [tex]$\square$[/tex] & [tex]$\square$[/tex] & \\
\hline
\end{tabular}



Answer :

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}

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