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: two or more even numbers
event b: an odd number on each of the first two rolls
event c: an even number on the last roll or the second roll (or both)

Explanation:

Step1: Determine the total number of outcomes

When rolling a number cube (with 6 faces) three times, the total number of possible outcomes is \(6\times6\times6 = 216\). But since we are dealing with odd - even parity (each roll has 2 possibilities: odd or even), the total number of outcomes in terms of odd - even combinations is \(2\times2\times2=8\) (as each roll has 2 parity states).

Step2: Analyze the event \(E = EOE\)

For the outcome \(EOE\):

• The probability of getting an even number on the first roll: There are 3 even numbers (\(2,4,6\)) out of 6 on a number cube. So the probability of an even number \(P(E_1)=\frac{3}{6}=\frac{1}{2}\).
• The probability of getting an odd number on the second roll: There are 3 odd numbers (\(1,3,5\)) out of 6. So the probability of an odd number \(P(O_2)=\frac{3}{6}=\frac{1}{2}\).
• The probability of getting an even number on the third roll: \(P(E_3)=\frac{3}{6}=\frac{1}{2}\).

Since the rolls are independent events, the probability of the combined event \(P(EOE)=P(E_1)\times P(O_2)\times P(E_3)\).

Step3: Calculate the probability

Substitute the values into the formula:

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Answer:

\(\frac{1}{8}\)