Understanding Dice Probability
When you throw two fair six-sided dice, each die has 6 possible outcomes ($1, 2, 3, 4, 5, 6$). To find the probability of a specific event, we use the formula:
$$P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$$
Step 1: Calculate Total Possible Outcomes
Since each die is independent, we multiply the number of outcomes for each die: $$6 \times 6 = 36$$ There are 36 total possible pairs $(d_1, d_2)$.
Step 2: Identify Favorable Outcomes
We need the sum $S = d_1 + d_2$ to be $\le 5$. Let's list the combinations systematically:
- Sum = 2: $(1, 1)$ — (1 outcome)
- Sum = 3: $(1, 2), (2, 1)$ — (2 outcomes)
- Sum = 4: $(1, 3), (2, 2), (3, 1)$ — (3 outcomes)
- Sum = 5: $(1, 4), (2, 3), (3, 2), (4, 1)$ — (4 outcomes)
Step 3: Calculate Total Favorable Cases
Summing these up: $1 + 2 + 3 + 4 = 10$ favorable outcomes.
Step 4: Final Calculation
$$P(S \le 5) = \frac{10}{36}$$ Reducing the fraction by dividing both numerator and denominator by 2, we get: $$P(S \le 5) = \frac{5}{18}$$
Intuition
Think of the outcomes as a grid. The sums increase as you move diagonally from the top-left $(1,1)$ towards the bottom-right $(6,6)$. Outcomes satisfying the condition form a small triangle in the top-left corner of the $6 \times 6$ grid.