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 event. Outcomes Probability ------------------------------ Event A: Two or more odd numbers Event B: Exactly one odd number ? Event C: An even number on the first roll

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. Outcomes Probability ------------------------------ Event A: Two or more odd numbers Event B: Exactly one odd number ? Event C: An even number on the first roll
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 event. \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|} \hline \multirow[t]{2}{*}{} & \multicolumn{8}{|c|}{Outcomes} & \multirow{2}{*}{Probability} \\ \hline & EEE & OEE & EEO & 000 & EOO & OEO & OOE & EOE & \\ \hline Event A: Two or more odd numbers & $\square$ & $\square$ & $\square$ & $\square$ & $\square$ & $\square$ & $\square$ & $\square$ & $\square$ \\ \hline Event B: Exactly one odd number & $\square$ & $\square$ & $\square$ & $\square$ & $\square$ & $\square$ & $\square$ & $\square$ & $?$ \\ \hline Event C: An even number on the first roll & $\square$ & $\square$ & $\square$ & $\square$ & $\square$ & $\square$ & $\square$ & $\square$ & $\square$ \\ \hline \end{tabular}
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Solution

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Solution Steps

To solve this problem, we need to identify the outcomes that match each event and then calculate the probability of each event. The probability of each outcome is equal since the number cube is fair.

  1. Event A: Two or more odd numbers - Identify outcomes with at least two 'O's.
  2. Event B: Exactly one odd number - Identify outcomes with exactly one 'O'.
  3. Event C: An even number on the first roll - Identify outcomes starting with 'E'.

The probability of each event is the number of favorable outcomes divided by the total number of outcomes (which is 8).

Step 1: Identify Outcomes for Event A

Event A requires two or more odd numbers. The outcomes that satisfy this condition are: \[ \text{OOO, EOO, OEO, OOE} \]

Step 2: Calculate Probability for Event A

The probability of Event A is the number of favorable outcomes divided by the total number of outcomes: \[ P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{4}{8} = \frac{1}{2} \]

Step 3: Identify Outcomes for Event B

Event B requires exactly one odd number. The outcomes that satisfy this condition are: \[ \text{OEE, EEO, EOE} \]

Step 4: Calculate Probability for Event B

The probability of Event B is the number of favorable outcomes divided by the total number of outcomes: \[ P(B) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{3}{8} \]

Step 5: Identify Outcomes for Event C

Event C requires an even number on the first roll. The outcomes that satisfy this condition are: \[ \text{EEE, EEO, EOO, EOE} \]

Step 6: Calculate Probability for Event C

The probability of Event C is the number of favorable outcomes divided by the total number of outcomes: \[ P(C) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{4}{8} = \frac{1}{2} \]

Final Answer

  • Event A outcomes: \(\text{OOO, EOO, OEO, OOE}\)
  • Probability of Event A: \(\boxed{\frac{1}{2}}\)
  • Event B outcomes: \(\text{OEE, EEO, EOE}\)
  • Probability of Event B: \(\boxed{\frac{3}{8}}\)
  • Event C outcomes: \(\text{EEE, EEO, EOO, EOE}\)
  • Probability of Event C: \(\boxed{\frac{1}{2}}\)
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