Question

c. is this game fair? explain your answer. (___ / 2 points)
d. what is the sum that has the highest probability of occurring when two dice are rolled? ____ (_ / 1 point)
what is the probability that the sum above will be rolled? ____ (_ / 1 point)

Explanation:

Part d (First Sub - Question: Sum with Highest Probability)

Step1: List all possible sums of two dice

When rolling two dice, each die has outcomes from 1 to 6. The possible sums \( s \) range from \( 1 + 1=2 \) to \( 6+6 = 12 \). We can list the number of ways to get each sum:

• Sum of 2: \((1,1)\) → 1 way.
• Sum of 3: \((1,2),(2,1)\) → 2 ways.
• Sum of 4: \((1,3),(2,2),(3,1)\) → 3 ways.
• Sum of 5: \((1,4),(2,3),(3,2),(4,1)\) → 4 ways.
• Sum of 6: \((1,5),(2,4),(3,3),(4,2),(5,1)\) → 5 ways.
• Sum of 7: \((1,6),(2,5),(3,4),(4,3),(5,2),(6,1)\) → 6 ways.
• Sum of 8: \((2,6),(3,5),(4,4),(5,3),(6,2)\) → 5 ways.
• Sum of 9: \((3,6),(4,5),(5,4),(6,3)\) → 4 ways.
• Sum of 10: \((4,6),(5,5),(6,4)\) → 3 ways.
• Sum of 11: \((5,6),(6,5)\) → 2 ways.
• Sum of 12: \((6,6)\) → 1 way.

Step2: Identify the sum with the most ways

From the above, the sum of 7 has 6 ways, which is more than the number of ways to get any other sum.

Step1: Calculate total number of possible outcomes

When rolling two dice, the total number of possible outcomes is \( 6\times6=36 \) (by the multiplication principle, as each die has 6 outcomes).

Step2: Calculate the probability

We know that the sum of 7 can occur in 6 ways. The probability \( P \) of an event is given by the formula \( P=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \). So for the sum of 7, the number of favorable outcomes is 6 and the total number of outcomes is 36. Thus, \( P=\frac{6}{36}=\frac{1}{6} \).

Answer:

7

Part d (Second Sub - Question: Probability of that Sum)