Simple Probability with Dice

Grade 5 Medium Probability & Statistics

Learning Objectives

  • List all outcomes when rolling one or two dice
  • Calculate the probability of simple events

Concept Explanation

Rolling dice is a classic way to understand probability. A fair six-sided die has 6 equally likely outcomes: 1, 2, 3, 4, 5, or 6. The probability of rolling a specific number is (\frac16). Students extend this by rolling two dice and exploring sums, where outcomes can vary (e.g., sum = 7 is more common than sum = 2).

This topic introduces the idea that some events are more likely than others based on the number of ways they can occur.

Worked Examples

Example 1

Problem: What is the probability of rolling a 4 on a fair 6-sided die?
Solution: (\frac16)
Explanation: There are 6 possible outcomes, and 4 is just one of them.

Example 2

Problem: List all outcomes for rolling one 6-sided die.
Solution: {1, 2, 3, 4, 5, 6}
Explanation: Each face is equally likely.

Example 3

Problem: You roll two dice. Which sum is more likely: 7 or 2?
Solution: 7 is more likely
Explanation: 7 can be formed by (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) → 6 ways, while 2 can be formed only by (1,1) → 1 way.

Common Errors

ErrorCorrectionReason
Assuming all sums are equally likelyCount the number of ways to form each sumDifferent sums have different combinations (e.g., 7 is the most common sum with two dice).
Mixing up theoretical probability and experimental resultsUnderstand probability is about likelihood, not certaintyActual trials can vary, but with many rolls, results tend to the theoretical probabilities.
Forgetting each die is independentEach roll doesn’t affect the nextRolling a 6 once doesn’t change the chances for the next roll.

Practice Problems

  1. Problem: What is the probability of rolling a 2 on a fair 6-sided die?
    Solution:

    (\frac1 6)

  2. Problem: List all outcomes for rolling a single die.
    Solution: 1, 2, 3, 4, 5, 6
  3. Problem: How many ways can you get a sum of 7 with two dice?
    Solution: 6 ways (1+6, 2+5, 3+4, 4+3, 5+2, 6+1)
  4. Problem: Which sum is impossible with one die?
    Solution: 7 (since a single die goes only up to 6)
  5. Problem: If you roll a die 10 times, will you always get each number once?
    Solution: No, probability is about chance, not certainty.

Real-World Application Example

When designing a board game, probability is crucial. You might want certain outcomes (like rolling a sum of 7) to be more common than others. Understanding dice probabilities ensures balanced and fun gameplay.

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