Question

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.
event a: an even number on both the first and the last rolls
event b: two or more even numbers
event c: an even number on the first roll or the third roll (or both)

Explanation:

Step1: Calculate probability for Event A

Event A: An even number on both the first and the last rolls.
Total number of outcomes \( n = 8 \).
Favorable outcomes for Event A: EEO, EEE. So \( m_A=2 \).
Probability \( P(A)=\frac{m_A}{n}=\frac{2}{8}=\frac{1}{4} \).

Step2: Calculate probability for Event B

Event B: Two or more even numbers.
Favorable outcomes for Event B: OEE, EEO, EOE, EOO, OOE, EEE. So \( m_B = 6 \).
Probability \( P(B)=\frac{m_B}{n}=\frac{6}{8}=\frac{3}{4} \).

Step3: Calculate probability for Event C

Event C: An even number on the first roll or the third roll (or both).
Favorable outcomes for Event C: OEO, OEE, EEO, EOE, EOO, OOE, EEE. So \( m_C=7 \).
Probability \( P(C)=\frac{m_C}{n}=\frac{7}{8} \).

Answer:

Event A: \(\frac{1}{4}\)
Event B: \(\frac{3}{4}\)
Event C: \(\frac{7}{8}\)