Two fair dice are thrown simultaneously. Find the probability that: a) the total is 7, b) the total is at least 8, c) the total is a prime number, d) at least one of the scores is... Two fair dice are thrown simultaneously. Find the probability that: a) the total is 7, b) the total is at least 8, c) the total is a prime number, d) at least one of the scores is a 6, e) exactly one of the scores is a 6, f) the two scores are the same, g) the difference between the scores is an odd number.
Understand the Problem
The question is asking for the probabilities of various outcomes when rolling two fair dice. Each part of the question pertains to different conditions regarding the totals and values shown on the dice.
Answer
a) $\frac{1}{6}$, b) $\frac{5}{12}$, c) $\frac{5}{12}$, d) $\frac{11}{36}$, e) $\frac{5}{18}$, f) $\frac{1}{6}$, g) $\frac{1}{2}$
Answer for screen readers
a) $P(\text{total is 7}) = \frac{1}{6}$
b) $P(\text{total is at least 8}) = \frac{5}{12}$
c) $P(\text{total is a prime number}) = \frac{5}{12}$
d) $P(\text{at least one score is a 6}) = \frac{11}{36}$
e) $P(\text{exactly one score is a 6}) = \frac{5}{18}$
f) $P(\text{two scores are the same}) = \frac{1}{6}$
g) $P(\text{difference is odd}) = \frac{1}{2}$
Steps to Solve
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Total number of outcomes Each die has 6 sides, so when two dice are rolled, the total number of possible outcomes is: $$ 6 \times 6 = 36 $$
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Probability the total is 7 The pairs that add up to 7 are: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). This gives us 6 favorable outcomes: $$ P(\text{total is 7}) = \frac{6}{36} = \frac{1}{6} $$
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Probability the total is at least 8 The pairs that add up to at least 8 are:
- Totals of 8: (2,6), (3,5), (4,4), (5,3), (6,2) → 5 outcomes
- Totals of 9: (3,6), (4,5), (5,4), (6,3) → 4 outcomes
- Totals of 10: (4,6), (5,5), (6,4) → 3 outcomes
- Totals of 11: (5,6), (6,5) → 2 outcomes
- Total of 12: (6,6) → 1 outcome
Adding these gives us: $$ 5 + 4 + 3 + 2 + 1 = 15 $$ Thus: $$ P(\text{total is at least 8}) = \frac{15}{36} = \frac{5}{12} $$
- Probability the total is a prime number The prime totals possible are 2, 3, 5, 7, 11. The favorable outcomes for each are:
- Total of 2: (1,1) → 1 outcome
- Total of 3: (1,2), (2,1) → 2 outcomes
- Total of 5: (1,4), (2,3), (3,2), (4,1) → 4 outcomes
- Total of 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) → 6 outcomes
- Total of 11: (5,6), (6,5) → 2 outcomes
Total favorable outcomes: $$ 1 + 2 + 4 + 6 + 2 = 15 $$ Thus: $$ P(\text{total is a prime number}) = \frac{15}{36} = \frac{5}{12} $$
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Probability at least one score is a 6 To find this, we can use complementary counting. First, calculate the outcomes where neither die shows a 6 (5 options per die): $$ 5 \times 5 = 25 $$ Therefore, the probability that at least one die is a 6 is: $$ P(\text{at least one 6}) = 1 - P(\text{no 6}) = 1 - \frac{25}{36} = \frac{11}{36} $$
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Probability exactly one score is a 6 The outcomes where exactly one die shows a 6 are: (6,1), (6,2), (6,3), (6,4), (6,5) and (1,6), (2,6), (3,6), (4,6), (5,6) → 10 outcomes total. Thus: $$ P(\text{exactly one 6}) = \frac{10}{36} = \frac{5}{18} $$
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Probability the two scores are the same The pairs where the scores are the same are: (1,1), (2,2), (3,3), (4,4), (5,5), (6,6) → 6 outcomes. Thus: $$ P(\text{two scores are the same}) = \frac{6}{36} = \frac{1}{6} $$
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Probability the difference between the scores is odd The difference is odd when one die shows an odd number and the other shows an even number. We have 3 odd numbers (1, 3, 5) and 3 even numbers (2, 4, 6). The pairs are: $$ 3 \times 3 + 3 \times 3 = 18 $$ Thus: $$ P(\text{odd difference}) = \frac{18}{36} = \frac{1}{2} $$
a) $P(\text{total is 7}) = \frac{1}{6}$
b) $P(\text{total is at least 8}) = \frac{5}{12}$
c) $P(\text{total is a prime number}) = \frac{5}{12}$
d) $P(\text{at least one score is a 6}) = \frac{11}{36}$
e) $P(\text{exactly one score is a 6}) = \frac{5}{18}$
f) $P(\text{two scores are the same}) = \frac{1}{6}$
g) $P(\text{difference is odd}) = \frac{1}{2}$
More Information
These outcomes are based on the fundamental principle of counting combinations of two dice rolls. The total number of outcomes (36) is derived from multiplying the number of sides on each die.
Tips
- Miscounting outcomes: Make sure to explicitly list the combinations or use systematic counting to avoid missing outcomes.
- Not using complementary probability: For problems involving "at least one," using the complementary probability can simplify counting.
- Forgetting to reduce fractions: Always reduce fractions to their simplest form when providing probabilities.
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