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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|}
\hline
Outcomes & OOE & EOE & OEO & EOO & EEO & OOO & EEE & OEE \\
\hline
Event A: An even number on the second roll & & & & & & & & \\
\hline
Event B: Two or more even numbers & & & & & & & & \\
\hline
\end{tabular}



Answer :

GinnyAnswer
Let's analyze the events and the outcomes step by step to fill in the table and calculate the probabilities for the events.

### Outcomes
Here are the 8 possible outcomes when rolling a number cube three times:
- OOE
- EOE
- OEO
- EOO
- EEO
- OOO
- EEE
- OEE

### Event A: An even number on the second roll
We need to check which outcomes have an even number (E) in the second position. Let's identify them:

- OOE: Second roll is O (odd), so it is not included.
- EOE: Second roll is O (odd), so it is not included.
- OEO: Second roll is E (even), so it is included.
- EOO: Second roll is O (odd), so it is not included.
- EEO: Second roll is E (even), so it is included.
- OOO: Second roll is O (odd), so it is not included.
- EEE: Second roll is E (even), so it is included.
- OEE: Second roll is E (even), so it is included.

The outcomes for Event A are:
OEO, EEO, EEE, OEE

Now we count the number of outcomes that satisfy this event. There are 4 such outcomes out of a total of 8. Therefore, the probability [tex]\( P(A) \)[/tex] is:
[tex]\[ P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{4}{8} = 0.5 \][/tex]

### Event B: Two or more even numbers
We need to check which outcomes have two or more even numbers (E). Let's identify them:

- OOE: One even number, so it is not included.
- EOE: One even number, so it is not included.
- OEO: Two even numbers, so it is included.
- EOO: One even number, so it is not included.
- EEO: Two even numbers, so it is included.
- OOO: Zero even numbers, so it is not included.
- EEE: Three even numbers, so it is included.
- OEE: Two even numbers, so it is included.

The outcomes for Event B are:
OEO, EEO, EEE, OEE

Now we count the number of outcomes that satisfy this event. There are 4 such outcomes out of a total of 8. Therefore, the probability [tex]\( P(B) \)[/tex] is:
[tex]\[ P(B) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{4}{8} = 0.5 \][/tex]

### Summary in Table Form

Let's fill in the table with the identified outcomes and probabilities.

[tex]\[ \begin{tabular}{|l|l|l|l|l|l|l|l|l|l|} \hline \multicolumn{2}{|l|}{} & \multicolumn{8}{|c|}{Outcomes} \\ \hline & OOE & EOE & OEO & EOO & EEO & OOO & EEE & OEE & Prob \\ \hline \begin{tabular}{l} Event A: An even number on the \\ second roll \end{tabular} & & & \checkmark & & \checkmark & & \checkmark & \checkmark & 0.5 \\ \hline \begin{tabular}{l} Event B: Two or more even numbers \end{tabular} & & & \checkmark & & \checkmark & & \checkmark & \checkmark & 0.5 \\ \hline \end{tabular} \][/tex]

This table summarizes the favorable outcomes for each event and provides the calculated probabilities.

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