Mathematics 9709/63 — May/June 2017

Walkthrough

We need the probability that the sum of two throws is 9.

First, the probabilities of faces 1, 2, 3, 4 and 5 are given, but face 6 is missing. Since the probabilities of all possible outcomes must add to 1, we find:

$$P(6) = 1 - (0.1+0.1+0.1+0.2+0.2) = 0.3$$

Next, because the two throws are independent, the probability of a particular ordered pair $(a,b)$ is $P(a) \times P(b)$.

The sum is 9 for the ordered pairs $(3,6)$, $(4,5)$, $(5,4)$ and $(6,3)$. Notice that both orders count: $(3,6)$ and $(6,3)$ are different outcomes, as are $(4,5)$ and $(5,4)$.

Now multiply the individual probabilities:

$$P(3,6)=0.1 \times 0.3 = 0.03$$ $$P(4,5)=0.2 \times 0.1 = 0.02$$ $$P(5,4)=0.1 \times 0.2 = 0.02$$ $$P(6,3)=0.3 \times 0.1 = 0.03$$

These four outcomes cannot happen at the same time, so they are mutually exclusive and we add their probabilities:

$$0.03 + 0.02 + 0.02 + 0.03 = 0.1$$

This can also be written as $(0.03 + 0.02) \times 2 = 0.1$.

Key Takeaways

• The total probability over all outcomes is 1, so a missing probability can be found by subtracting the known probabilities from 1.
• Two throws are independent, so multiply the individual probabilities for an ordered pair.
• Order matters when the two throws are distinguishable, so $(3,6)$ and $(6,3)$ are both counted.
• Mutually exclusive outcomes are combined by addition.

Common Mistakes

• Forgetting to find $P(6)$ first.
• Treating all faces as equally likely; the die is biased, so face probabilities are not all $\frac{1}{6}$.
• Missing one of the ordered pairs, especially $(3,6)$ and $(6,3)$ or $(4,5)$ and $(5,4)$.
• Multiplying probabilities but forgetting to add the mutually exclusive cases.
• Using $P(5)=0.2$ instead of $P(5)=0.1$.

Things to Be Careful About

• The known probabilities are $3 \times 0.1 + 2 \times 0.2 = 0.7$, so $P(6)=0.3$.
• Face 4 has probability 0.2 and face 5 has probability 0.1; do not swap these.
• Both orders must be included for each valid pair.
• The final answer should be simplified to 0.1, though unsimplified products are accepted in the working.