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A number cube is rolled three times. An outcome is represented by a string of the sort OEO (meaning an odd number on the first roll, an even number on the second roll, and an odd 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}{|l|c|c|c|c|c|c|c|c|}
\hline
& 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 carefully analyze and determine the outcomes for each event and then calculate the respective probabilities.

### Step-by-Step Solution:

#### Outcomes:
First, let's list the eight possible outcomes when a number cube is rolled three times:

1. OOE - Odd, Odd, Even
2. EOE - Even, Odd, Even
3. OEO - Odd, Even, Odd
4. EOO - Even, Odd, Odd
5. EEO - Even, Even, Odd
6. OOO - Odd, Odd, Odd
7. EEE - Even, Even, Even
8. OEE - Odd, Even, Even

Each of these outcomes has an equal probability of occurring since each roll is independent and has the same probability.

### Event Analysis:
#### Event A: An even number on the second roll
For this event, we list outcomes where the second roll is an even number:
- EOE
- OEO
- EEO
- OEE

Probability of Event A:
[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
For this event, we list outcomes where there are at least two even numbers:
- EOE (Even, Odd, Even)
- EEO (Even, Even, Odd)
- EEE (Even, Even, Even)
- OEE (Odd, Even, Even)

Probability of Event B:
[tex]\[ P(B) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{4}{8} = 0.5 \][/tex]

### Event Distribution Table:

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

In summary:
- Event A (An even number on the second roll) involves the outcomes: EOE, OEO, EEO, OEE
- Event B (Two or more even numbers) involves the outcomes: EOE, EEO, EEE, OEE
- Both events have a probability of [tex]\(0.5\)[/tex].

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