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

Sagot :

GinnyAnsweridn
We have a scenario where a number cube (6-sided die) is rolled three times. Each possible outcome is represented by a string of odd (O) and even (E) values corresponding to the results of each roll. For example, OEE indicates that the first roll was odd, the second roll was even, and the third roll was even. There are 8 possible outcomes, each equally likely.

Here is the table of outcomes:
- OEO
- EOE
- EOO
- OOO
- EEO
- OEE
- OOE
- EEE

We need to determine three different events and calculate the probability of each event.

### Event A: An even number on both the first and the last rolls
To satisfy this event, we need the first and third rolls to be even. The outcomes that match this condition are:
- EOE
- EEE

Therefore, there are 2 favorable outcomes. Since there are 8 possible outcomes, the probability of Event A is:
[tex]\[ \text{Probability of Event A} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{2}{8} = 0.25 \][/tex]

### Event B: An odd number on each of the last two rolls
To satisfy this event, we need the second and third rolls to be odd. The outcomes that match this condition are:
- EOO
- OOO

Here, we again have 2 favorable outcomes out of 8 possible outcomes. Thus, the probability of Event B is:
[tex]\[ \text{Probability of Event B} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{2}{8} = 0.25 \][/tex]

### Event C: Two or more odd numbers
To satisfy this event, we need at least two of the rolls to be odd. The outcomes that meet this condition are:
- OEO
- EOO
- OOO
- OOE

In this case, there are 4 favorable outcomes out of 8 possible outcomes. Hence, the probability of Event C is:
[tex]\[ \text{Probability of Event C} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{4}{8} = 0.5 \][/tex]

In summary, the events and their probabilities are:
- Event A (Even number on both the first and the last rolls): [tex]\( 0.25 \)[/tex]
- Event B (Odd number on each of the last two rolls): [tex]\( 0.25 \)[/tex]
- Event C (Two or more odd numbers): [tex]\( 0.5 \)[/tex]
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