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 odd number on each of the first two rolls	✓	✓							$\frac{1}{4}$
event b: an even number on the last roll		✓	✓		✓		✓		$\frac{1}{2}$
event c: two or more odd numbers	✓	✓						✓	$\frac{5}{8}$

Explanation:

Step1: Determine total number of outcomes

A number - cube is rolled 3 times. Each roll has 2 possibilities (even or odd), so the total number of outcomes is $2\times2\times2 = 8$ since each roll is an independent event.

Step2: Analyze Event A

Event A is an odd number on each of the first two rolls. The favorable outcomes are OOO and OOE. The number of favorable outcomes $n(A)=2$. The probability $P(A)=\frac{n(A)}{n(S)}=\frac{2}{8}=\frac{1}{4}$, where $n(S) = 8$ is the total number of outcomes.

Step3: Analyze Event B

Event B is an even number on the last roll. The favorable outcomes are OOE, EEE, EEO, OEE. The number of favorable outcomes $n(B)=4$. The probability $P(B)=\frac{n(B)}{n(S)}=\frac{4}{8}=\frac{1}{2}$.

Step4: Analyze Event C

Event C is two or more odd numbers. The favorable outcomes are OOO, OOE, OEO, EOO, EOE. The number of favorable outcomes $n(C)=5$. The probability $P(C)=\frac{n(C)}{n(S)}=\frac{5}{8}$.

Answer:

Event A: Check OOO, OOE; Probability: $\frac{1}{4}$
Event B: Check OOE, EEE, EEO, OEE; Probability: $\frac{1}{2}$
Event C: Check OOO, OOE, OEO, EOO, EOE; Probability: $\frac{5}{8}$