Probability distribution of non-negative differences from rolling two six-sided dice.
Check the final answer first, then review the worked steps.
Problem
Probability distribution of non-negative differences from rolling two six-sided dice.
Answer
Frequency for difference 2 is 110, Probability for difference 2 is 0.22
Step-by-step solution
1. Calculate the missing frequency: The total number of rolls simulated is 500. We are given the frequencies for differences 0, 1, 3, 4, and 5. To find the frequency for a difference of 2, we subtract the sum of the known frequencies from the total number of rolls.
Known frequencies: 88 (for difference 0) + 134 (for difference 1) + 86 (for difference 3) + 56 (for difference 4) + 26 (for difference 5) = 390.
Missing frequency = Total rolls - Sum of known frequencies = $500 - 390 = 110$.
2. Calculate the missing probability: The probability of an event is calculated by dividing the frequency of that event by the total number of trials (simulated rolls). We have already calculated the frequency for a difference of 2 to be 110, and the total number of rolls is 500.
Probability (Difference = 2) = Frequency (Difference = 2) / Total rolls = $110 / 500 = 0.22$.
3. Verify the probabilities sum to 1: The sum of all probabilities in a probability distribution must equal 1. Let's check this with the given and calculated probabilities.
Given probabilities: 0.176 (for difference 0) + 0.268 (for difference 1) + 0.172 (for difference 3) + 0.112 (for difference 4) + 0.052 (for difference 5) = 0.78.
Calculated probability for difference 2 = 0.22.
Total sum of probabilities = $0.78 + 0.22 = 1.00$.
This confirms our calculations are correct.