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: alternating odd number and even number (with either coming first)
event b: an even number on the first roll
event c: exactly one odd number
outcomes
eeo
ooe
eee
eoe
eoo
ooo
oee
oeo
probability

Explanation:

Event A: Alternating odd number and even number (with either coming first)

Step1: Analyze each outcome

• EEO: Not alternating (two evens first)
• OOE: Not alternating (two odds first)
• EEE: Not alternating (all evens)
• EOE: Alternating (E, O, E)
• EOO: Not alternating (E, O, O)
• OOO: Not alternating (all odds)
• OEE: Alternating (O, E, E)
• OEO: Alternating (O, E, O)

Step2: Calculate probability

There are 8 total outcomes. The favorable outcomes for Event A (alternating) are EOE, OEE, OEO. So the number of favorable outcomes \(n = 3\).
The probability \(P(A)=\frac{n}{N}\), where \(N = 8\). So \(P(A)=\frac{3}{8}\)

Event B: An even number on the first roll

Step1: Analyze each outcome

• EEO: First roll is even
• OOE: First roll is odd
• EEE: First roll is even
• EOE: First roll is even
• EOO: First roll is even
• OOO: First roll is odd
• OEE: First roll is odd
• OEO: First roll is odd

Step2: Calculate probability

The favorable outcomes for Event B (first roll even) are EEO, EEE, EOE, EOO. So \(n = 4\).
The probability \(P(B)=\frac{n}{N}=\frac{4}{8}=\frac{1}{2}\)

Event C: Exactly one odd number

Step1: Analyze each outcome

• EEO: One odd number
• OOE: Two odd numbers
• EEE: Zero odd numbers
• EOE: One odd number
• EOO: Two odd numbers
• OOO: Three odd numbers
• OEE: One odd number
• OEO: Two odd numbers

Step2: Calculate probability

The favorable outcomes for Event C (exactly one odd) are EEO, EOE, OEE. So \(n = 3\).
The probability \(P(C)=\frac{n}{N}=\frac{3}{8}\)

Answer:

For Event A: \(\frac{3}{8}\)
For Event B: \(\frac{1}{2}\)
For Event C: \(\frac{3}{8}\)