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Antonio rolls two fair six-sided dice and calculates the difference between the scores - OCR - GCSE Maths - Question 12 - 2019 - Paper 6
Question 12
Antonio rolls two fair six-sided dice and calculates the difference between the scores. For example, if the two scores are 2 and 5 or 5 and 2 then the difference is ... show full transcript
Worked Solution & Example Answer:Antonio rolls two fair six-sided dice and calculates the difference between the scores - OCR - GCSE Maths - Question 12 - 2019 - Paper 6
Step 1
Complete the sample space diagram to show the possible outcomes from Antonio's dice.
Answer
To complete the sample space diagram for the difference between the scores when rolling two dice, we fill in the table based on the absolute differences:
-
When Dice 1 shows 1:
- Difference with 1: 0
- Difference with 2: 1
- Difference with 3: 2
- Difference with 4: 3
- Difference with 5: 4
- Difference with 6: 5
-
When Dice 1 shows 2:
- Difference with 1: 1
- Difference with 2: 0
- Difference with 3: 1
- Difference with 4: 2
- Difference with 5: 3
- Difference with 6: 4
-
When Dice 1 shows 3:
- Difference with 1: 2
- Difference with 2: 1
- Difference with 3: 0
- Difference with 4: 1
- Difference with 5: 2
- Difference with 6: 3
-
When Dice 1 shows 4:
- Difference with 1: 3
- Difference with 2: 2
- Difference with 3: 1
- Difference with 4: 0
- Difference with 5: 1
- Difference with 6: 2
-
When Dice 1 shows 5:
- Difference with 1: 4
- Difference with 2: 3
- Difference with 3: 2
- Difference with 4: 1
- Difference with 5: 0
- Difference with 6: 1
-
When Dice 1 shows 6:
- Difference with 1: 5
- Difference with 2: 4
- Difference with 3: 3
- Difference with 4: 2
- Difference with 5: 1
- Difference with 6: 0
With this, all entries in the table can be filled in properly.
Step 2
Calculate the probability that he gets a difference of 1 on all three rolls.
Answer
To calculate the probability of rolling a difference of 1 on all three rolls:
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Identify successful outcomes: To find the outcomes that yield a difference of 1, we note the following results from the completed table:
- (1, 2), (2, 1)
- (2, 3), (3, 2)
- (3, 4), (4, 3)
- (4, 5), (5, 4)
- (5, 6), (6, 5)
This totals 10 successful outcomes.
-
Identify total outcomes: The total number of rolls when rolling two dice 3 times is:
-
Calculate probability: The probability of getting a difference of 1 on one roll is: rac{10}{36} = rac{5}{18}
Therefore, for three rolls, the probability of getting a difference of 1 each time is: rac{5}{18} imes rac{5}{18} imes rac{5}{18} = rac{125}{5832}
Thus, the answer is: