6. what are the odds in favour of rolling each of the following totals with a standard pair of dice?\na)…

Explanation:

Step1: Find total number of outcomes

When rolling two dice, each die has 6 possible outcomes. So the total number of outcomes is $6\times6 = 36$.

Step2: Calculate odds for part a (total of 7)

The combinations that give a sum of 7 are (1,6), (2,5), (3,4), (4,3), (5,2), (6,1), so there are 6 favorable outcomes. The number of non - favorable outcomes is $36 - 6=30$. The odds in favor are $\frac{6}{30}=\frac{1}{5}$, or 1:5.

Step3: Calculate odds for part b (doubles)

The doubles are (1,1), (2,2), (3,3), (4,4), (5,5), (6,6), so there are 6 favorable outcomes. The number of non - favorable outcomes is $36 - 6 = 30$. The odds in favor are $\frac{6}{30}=\frac{1}{5}$, or 1:5.

Step4: Calculate odds for part c (less than 4)

The sums less than 4 are 2 and 3. For a sum of 2: (1,1); for a sum of 3: (1,2), (2,1). So there are 3 favorable outcomes. The number of non - favorable outcomes is $36 - 3=33$. The odds in favor are $\frac{3}{33}=\frac{1}{11}$, or 1:11.

Step5: Calculate odds for part d (greater than 4)

The sums greater than 4 are 5, 6, 7, 8, 9, 10, 11, 12. The number of sums that are 4 or less are 3 (sums of 2 and 3 as calculated above). So there are $36 - 3 = 33$ favorable outcomes and 3 non - favorable outcomes. The odds in favor are $\frac{33}{3}=11$, or 11:1.

Answer:

a) 1:5 b) 1:5 c) 1:11 d) 11:1