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

Explanation:

Step1: Analyze Event A (More even than odd)

A valid outcome for Event A must have 2 or 3 even numbers (since 3 rolls: 2 even,1 odd or 3 even,0 odd). Let's check each outcome:

• EOO: 1 even (E), 2 odd (O) → No
• OEE: 2 even (E), 1 odd (O) → Yes
• EOE: 2 even (E), 1 odd (O) → Yes
• OOO: 0 even → No
• OOE: 1 even → No
• OEO: 1 even → No
• EEE: 3 even → Yes
• EEO: 2 even → Yes

So mark OEE, EOE, EEE, EEO for Event A.

Step2: Analyze Event B (Alternating even-odd, first odd)

The pattern should be Odd - Even - Odd (since first is odd, then even, then odd) or Odd - Even - Even? Wait, "Alternating even number and odd number (with either coming first)"? Wait, the description: "Alternating even number and odd number (with either coming first)". Wait, first is odd, then even, then odd? Or odd - even - even? Wait, alternating means switching. So if first is odd, second even, third odd (OEO) or odd - even - even? No, alternating: odd, even, odd (OEO) or even, odd, even (EOE). Wait, the event says "with either coming first" → so first odd: O - E - O (OEO) or first even: E - O - E (EOE). Wait, let's re - read: "Alternating even number and odd number (with either coming first)". So possible patterns: O - E - O (starts with odd, alternates) or E - O - E (starts with even, alternates). Let's check each outcome:

• EOO: E - O - O → not alternating (last two O) → No
• OEE: O - E - E → not alternating (last two E) → No
• EOE: E - O - E → alternating (E - O - E) → Yes
• OOO: O - O - O → No
• OOE: O - O - E → No
• OEO: O - E - O → alternating (O - E - O) → Yes
• EEE: E - E - E → No
• EEO: E - E - O → No

So mark EOE, OEO for Event B.

Step3: Analyze Event C (Even on first roll)

Outcomes with first roll E:

• EOO: First E → Yes
• OEE: First O → No
• EOE: First E → Yes
• OOO: First O → No
• OOE: First O → No
• OEO: First O → No
• EEE: First E → Yes
• EEO: First E → Yes

So mark EOO, EOE, EEE, EEO for Event C.

Step4: Calculate Probability (assuming each outcome has equal probability, total 8 outcomes)

For Event A: Number of favorable outcomes = 4 (OEE, EOE, EEE, EEO) → Probability = $\frac{4}{8}=\frac{1}{2}$
For Event B: Number of favorable outcomes = 2 (EOE, OEO) → Probability = $\frac{2}{8}=\frac{1}{4}$
For Event C: Number of favorable outcomes = 4 (EOO, EOE, EEE, EEO) → Probability = $\frac{4}{8}=\frac{1}{2}$

(Note: The problem might ask for one event's probability, but since it's about marking and probability, we'll assume the final answer for probability (e.g., for Event A: 4/8 = 1/2, Event B: 2/8 = 1/4, Event C: 4/8 = 1/2))

Answer:

For Event A: Probability $\frac{1}{2}$; For Event B: Probability $\frac{1}{4}$; For Event C: Probability $\frac{1}{2}$ (depending on the event, but following the steps above)