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.

	outcomes	probability
event a: an even number on the second roll or the third roll (or both)
event b: alternating even number and odd number (with either coming first)
event c: no even numbers on the first two rolls

Explanation:

Step1: Recall probability formula

Probability of an event $P(E)=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$. Here, the total number of outcomes is $n = 8$.

Step2: Analyze Event A

Event A is an even number on the second roll or the third roll (or both).
Favorable outcomes for Event A: EOE, OOE, EEE, OEE, EEO, OEO. So the number of favorable outcomes $n_A=6$.
$P(A)=\frac{6}{8}=\frac{3}{4}$.

Step3: Analyze Event B

Event B is alternating even - number and odd - number (with either coming first).
Favorable outcomes for Event B: EOE, OEO. So the number of favorable outcomes $n_B = 2$.
$P(B)=\frac{2}{8}=\frac{1}{4}$.

Step4: Analyze Event C

Event C is no even numbers on the first two rolls.
Favorable outcomes for Event C: OOO, OOE. So the number of favorable outcomes $n_C=2$.
$P(C)=\frac{2}{8}=\frac{1}{4}$.

Answer:

Event	Outcomes	Probability
Event B	EOE, OEO	$\frac{1}{4}$
Event C	OOO, OOE	$\frac{1}{4}$