Math Problem Statement

A number cube is rolled three times. For each event, check the outcomes that are contained in the event and enter the probability of the event.

Solution

To solve the problem, we must determine which outcomes satisfy each event and calculate the probability.

Events and Steps:

  1. Outcomes: The possible outcomes are:

    • EOO, OEO, EOE, EEE, EEE, OOE, OOO, EEO
      (8 outcomes total, where each outcome represents even/odd results across 3 rolls.)
  2. Probability of Each Outcome: Since all outcomes are equally likely, the probability of each outcome is: P(each outcome)=18P(\text{each outcome}) = \frac{1}{8}


Event A: An even number on the last roll or the second roll (or both).

  • Focus on outcomes where the second roll or last roll (or both) is even.
  • Outcomes that satisfy Event A:
    • EOO, OEO, EOE, EEE, EEO
  • Total outcomes = 5
  • Probability: P(Event A)=58P(\text{Event A}) = \frac{5}{8}

Event B: More even numbers than odd numbers.

  • An outcome has more even numbers if at least 2 positions are even (e.g., EEE, EEO, etc.).
  • Outcomes that satisfy Event B:
    • EOE, EEE, EEO
  • Total outcomes = 3
  • Probability: P(Event B)=38P(\text{Event B}) = \frac{3}{8}

Event C: Two or more even numbers.

  • Focus on outcomes with at least 2 even numbers.
  • Outcomes that satisfy Event C:
    • EOE, EEE, EEO
  • Total outcomes = 3
  • Probability: P(Event C)=38P(\text{Event C}) = \frac{3}{8}

Final Table:

EventOutcomes SatisfiedProbability
Event AEOO, OEO, EOE, EEE, EEO58\frac{5}{8}
Event BEOE, EEE, EEO38\frac{3}{8}
Event CEOE, EEE, EEO38\frac{3}{8}

Would you like a detailed explanation for any specific event or further clarifications?

5 Related Questions:

  1. How do you calculate probabilities for equally likely outcomes?
  2. What are the differences between "at least two" and "more than two" conditions?
  3. How would probabilities change if we added a fourth roll?
  4. Can you explain complementary events and how they relate to these probabilities?
  5. How do we use probability in real-world situations like dice games?

Tip: Always verify the outcomes carefully to ensure none are missed when counting for specific events.

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Math Problem Analysis

Mathematical Concepts

Probability
Equally likely outcomes
Events and outcomes

Formulas

P(each outcome) = 1/8

Theorems

Basic probability theorem
Equally likely outcomes principle

Suitable Grade Level

Grades 6-8