Probability of rolling at least a 7 with two dice

Question

What is the probability of rolling at least a 7 when given two dice?

Answer 1

If a 1 is rolled with the first dice, only rolling a 6 on the second will lead to 7.

If a 2 is rolled with the first dice, rolling a 5 or a 6 on the second will lead to 7.

This pattern continues...

P( rolling at least 7 ) = sum of all ( P( first dice ) * P( second dice ) )

P = (1/6 * 1/6) + (1/6 * 2/6) + (1/6 * 3/6) + (1/6 * 4/6) + (1/6 * 5/6) + (1/6 * 6/6) = 3.5/6 = .5833.

Thus, there is a 58.3% probability of rolling at least a 7 when given two dice.

Answer 2

Total outcomes of rolling two dice = 6*6 = 36

Different combinations possible with the rolling of 2 dice-
(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),(5,1),(5,2),(5,3),(5,4),(5,5),(5,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)

Favorable outcomes = (1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,3),(4,4),(4,5),(4,6),(5,2),(5,3),(5,4),(5,5),(5,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)

Probability = Favorable outcomes/Total outcomes * 100
                 = (21/36)*100
                 = 58.3%

Answer 3

1. Each die has 6 faces, so there are 6 * 6 = 36 possible outcomes when rolling two dice.

2. If the sum of two dice is at least a 7, the sum can be 7, 8, 9, 10, 11, or 12.

• Sum of the dice = 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) - 6 outcomes
• Sum of the dice = 8: (2,6), (3,5), (4,4), (5,3), (6,2) - 5 outcomes
• Sum of the dice = 9: (3,6), (4,5), (5,4), (6,3) - 4 outcomes
• Sum of the dice = 10: (4,6), (5,5), (6,4) - 3 outcomes
• Sum of the dice = 11: (5,6), (6,5) - 2 outcomes
• Sum of the dice = 12: (6,6) - 1 outcome

3. Add all possible outcomes: 6 + 5 + 4 + 3 + 2 + 1 = 21

4. The probability of rolling at least a 7 when given two dice:

    21 / 36 = 7 / 12 ≈ 0.5833