Question

refer to the table below. of the 36 possible outcomes, determine the number for which the sum (for both dice) is greater than 8.
die 1 \\ die 2 1 2 3 4 5 6
1 (1,1) (1,2) (1,3) (1,4) (1,5) (1,6)
2 (2,1) (2,2) (2,3) (2,4) (2,5) (2,6)
3 (3,1) (3,2) (3,3) (3,4) (3,5) (3,6)
4 (4,1) (4,2) (4,3) (4,4) (4,5) (4,6)
5 (5,1) (5,2) (5,3) (5,4) (5,5) (5,6)
6 (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)
there are \\(\square\\) ways that the sum can be greater than 8.

Explanation:

Step1: Identify sums >8

We need to find all pairs \((a,b)\) where \(a + b>8\), \(a\) is Die 1, \(b\) is Die 2.

Step2: Check each row

- Row 1 (Die 1 = 1): \(1 + b>8\Rightarrow b > 7\), but \(b\leq6\), so 0 outcomes.
- Row 2 (Die 1 = 2): \(2 + b>8\Rightarrow b > 6\), \(b\leq6\), so 0 outcomes.
- Row 3 (Die 1 = 3): \(3 + b>8\Rightarrow b > 5\), so \(b = 6\). Outcome: \((3,6)\). 1 outcome.
- Row 4 (Die 1 = 4): \(4 + b>8\Rightarrow b > 4\), so \(b = 5,6\). Outcomes: \((4,5),(4,6)\). 2 outcomes.
- Row 5 (Die 1 = 5): \(5 + b>8\Rightarrow b > 3\), so \(b = 4,5,6\). Outcomes: \((5,4),(5,5),(5,6)\). 3 outcomes.
- Row 6 (Die 1 = 6): \(6 + b>8\Rightarrow b > 2\), so \(b = 3,4,5,6\). Outcomes: \((6,3),(6,4),(6,5),(6,6)\). 4 outcomes.

Step3: Sum the outcomes

Total outcomes: \(0 + 0+1 + 2+3 + 4=10\)? Wait, no, wait: Wait, let's recalculate. Wait, when Die 1 is 3: \(3 + 6 = 9>8\), correct (1). Die 1=4: \(4 + 5=9\), \(4 + 6 = 10\) (2). Die 1=5: \(5 + 4=9\), \(5 + 5 = 10\), \(5 + 6 = 11\) (3). Die 1=6: \(6 + 3=9\), \(6 + 4 = 10\), \(6 + 5 = 11\), \(6 + 6 = 12\) (4). Wait, also, check Die 1=3: \(3+6=9\). Die 1=4: \(4+5=9\), \(4+6=10\). Die 1=5: \(5+4=9\), \(5+5=10\), \(5+6=11\). Die 1=6: \(6+3=9\), \(6+4=10\), \(6+5=11\), \(6+6=12\). Also, what about Die 1=3, Die 2=6 (sum 9); Die 1=4, Die 2=5 (9), Die 2=6 (10); Die 1=5, Die 2=4 (9), Die 2=5 (10), Die 2=6 (11); Die 1=6, Die 2=3 (9), Die 2=4 (10), Die 2=5 (11), Die 2=6 (12). Wait, also, did we miss Die 1=3, Die 2=6 (sum 9); Die 1=4, Die 2=5 (9), Die 2=6 (10); Die 1=5, Die 2=4 (9), Die 2=5 (10), Die 2=6 (11); Die 1=6, Die 2=3 (9), Die 2=4 (10), Die 2=5 (11), Die 2=6 (12). Wait, also, let's list all pairs:

From the table:

- Die 1=3: (3,6) → sum 9
- Die 1=4: (4,5), (4,6) → sums 9,10
- Die 1=5: (5,4), (5,5), (5,6) → sums 9,10,11
- Die 1=6: (6,3), (6,4), (6,5), (6,6) → sums 9,10,11,12

Wait, also, what about Die 1=2: no, Die 1=1: no. Wait, another way: list all pairs where sum >8 (i.e., sum ≥9):

Possible sums: 9,10,11,12.

Sum=9: (3,6),(4,5),(5,4),(6,3) → 4 outcomes.

Sum=10: (4,6),(5,5),(6,4) → 3 outcomes.

Sum=11: (5,6),(6,5) → 2 outcomes.

Sum=12: (6,6) → 1 outcome.

Total: \(4 + 3+2 + 1=10\)? Wait, no: 4 (sum9) +3 (sum10)+2 (sum11)+1 (sum12)=10? Wait, no, let's count again:

Sum=9: (3,6), (4,5), (5,4), (6,3) → 4.

Sum=10: (4,6), (5,5), (6,4) → 3.

Sum=11: (5,6), (6,5) → 2.

Sum=12: (6,6) → 1.

Total: 4+3=7, 7+2=9, 9+1=10? Wait, no, wait: (3,6), (4,5), (5,4), (6,3) → 4; (4,6), (5,5), (6,4) → 3 (total 7); (5,6), (6,5) → 2 (total 9); (6,6) → 1 (total 10). Wait, but earlier when we checked each row:

Row 3 (Die1=3): 1 outcome (3,6)

Row4 (Die1=4): 2 outcomes (4,5),(4,6)

Row5 (Die1=5): 3 outcomes (5,4),(5,5),(5,6)

Row6 (Die1=6): 4 outcomes (6,3),(6,4),(6,5),(6,6)

Total: 1+2+3+4=10. Yes, that's correct. So the total number of outcomes where sum >8 is 10? Wait, no, wait: Wait, sum >8 means sum ≥9. Let's list all pairs:

From the table:

- (3,6): sum 9

- (4,5): sum 9; (4,6): sum 10

- (5,4): sum 9; (5,5): sum 10; (5,6): sum 11

- (6,3): sum 9; (6,4): sum 10; (6,5): sum 11; (6,6): sum 12

Now count them: 1 (from row3) + 2 (row4) + 3 (row5) + 4 (row6) = 10. Wait, but let's count the actual pairs:

(3,6), (4,5), (4,6), (5,4), (5,5), (5,6), (6,3), (6,4), (6,5), (6,6). That's 10 pairs. So the number of ways is 10? Wait, no, wait, I think I made a mistake earlier. Wait, let's check sum >8: sum is greater than 8, so sum ≥9. Let's list all possible (a,b) where a + b ≥9, a from 1 - 6, b from 1 - 6.

For a=1: 1 + b ≥9 → b ≥8 → impossible (b≤6).

For a=2: 2 + b ≥9 → b ≥7 → impossible.

For a=3: 3 + b ≥9 → b ≥6 → b=6 → (3,6). 1.

For a=4: 4 + b ≥9 → b ≥5 → b=5,6 → (4,5), (4,6). 2.

For a=5: 5 + b ≥9 → b ≥4 → b=4,5,6 → (5,4), (5,5), (5,6). 3.

For a=6: 6 + b ≥9 → b ≥3 → b=3,4,5,6 → (6,3), (6,4), (6,5), (6,6). 4.

Total: 1 + 2 + 3 + 4 = 10. Yes, that's correct.

Answer:

10