Questions: 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 Outcomes Probability EEO OOE OEO EOO EOE OEE 000 EEE Event A: An odd number on each of the last two rolls - 0 Event B: Two or more odd numbers 0 口 0 - - Event C: Exactly one odd number 0 0 0 0 0 0

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 Outcomes Probability EEO OOE OEO EOO EOE OEE 000 EEE Event A: An odd number on each of the last two rolls - 0 Event B: Two or more odd numbers 0 口 0 - - Event C: Exactly one odd number 0 0 0 0 0 0
Transcript text: 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 \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|} \hline & \multicolumn{8}{|c|}{Outcomes} & \multirow[b]{2}{*}{Probability} \\ \hline & EEO & OOE & OEO & EOO & EOE & OEE & 000 & EEE & \\ \hline Event A: An odd number on each of the last two rolls & $\square$ & $\square$ & $\square$ & $\square$ & $\square$ & $\square$ & - & 0 & $\square$ \\ \hline Event B: Two or more odd numbers & $\square$ & 0 & $\square$ & 口 & 0 & $\square$ & - & - & $\square$ \\ \hline Event C: Exactly one odd number & 0 & 0 & 0 & $\square$ & 0 & 0 & 0 & $\square$ & $\square$ \\ \hline \end{tabular}
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Solution

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Solution Steps

Step 1: Understanding the Problem

We are given a number cube (a six-sided die) that is rolled three times. Each roll can result in an odd or even number. The outcomes are represented by strings such as OEE, where 'O' stands for an odd number and 'E' stands for an even number. We need to determine which outcomes belong to each event and calculate the probability of each event.

Step 2: Analyzing Event A

Event A: An odd number on each of the last two rolls.
We need to find outcomes where the second and third rolls are odd. The possible outcomes are:

  • OOE
  • OOO

These are the only outcomes where the last two rolls are odd.

Step 3: Calculating Probability for Event A

There are 8 possible outcomes in total, and 2 of them satisfy Event A. Since each outcome is equally likely, the probability of Event A is:

\[ P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{2}{8} = 0.25 \]

Step 4: Analyzing Event B

Event B: Two or more odd numbers.
We need to find outcomes with at least two odd numbers. The possible outcomes are:

  • OOE
  • OEO
  • EOO
  • OEE
  • OOO

These outcomes have two or more odd numbers.

Step 5: Calculating Probability for Event B

There are 5 outcomes that satisfy Event B. Thus, the probability of Event B is:

\[ P(B) = \frac{5}{8} = 0.625 \]

Step 6: Analyzing Event C

Event C: Exactly one odd number.
We need to find outcomes with exactly one odd number. The possible outcomes are:

  • EOO
  • EEE

These outcomes have exactly one odd number.

Step 7: Calculating Probability for Event C

There are 2 outcomes that satisfy Event C. Thus, the probability of Event C is:

\[ P(C) = \frac{2}{8} = 0.25 \]

Final Answer

  • Event A: Outcomes are OOE, OOO. Probability is \(\boxed{0.25}\).
  • Event B: Outcomes are OOE, OEO, EOO, OEE, OOO. Probability is \(\boxed{0.625}\).
  • Event C: Outcomes are EOO, EEE. Probability is \(\boxed{0.25}\).
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