在阳光岛共和国，所有男性公民在入伍服兵役之前都必须进行国民体能评估（NPFA）测试。根据历史数据，参加 NPFA 测试的男性中有 75% 能通过该测试，并被授予铜奖、银奖或金奖标准。已知 NPFA 测试的表现会影响这些国民服役人员晋升为军官（Officer）的机会。 对于未能通过 NPFA 测试的男性公民，他们被要求在基础训练结束时进行体能评估测试（PAT）。已知通过 PAT 的人将有机会晋升为军官。该信息总结在下面的树状图中。

1. Set up an equation for the total probability of a randomly chosen serviceman being an Officer by summing the probabilities of all paths leading to 'Officer'.

$\text{P}(\text{Officer}) = 0.75(0.20 \times 0.91 + 0.63 \times 0.72 + 0.17 \times 0.54) + 0.25(a \times 0.36)$
$0.60 = 0.75(0.182 + 0.4536 + 0.0918) + 0.09a$
$0.60 = 0.54555 + 0.09a$

2. Solve the linear equation for $a$.

$0.09a = 0.60 - 0.54555$
$0.09a = 0.05445$
$a = \frac{0.05445}{0.09} = 0.605$

1. Calculate the probability that a serviceman fails the NPFA and becomes an Officer, using the value of $a$ found in part (i) ($a = 0.605$).

$P(\text{Fail NPFA} \cap \text{Officer}) = 0.25 \times a \times 0.36$
$P(\text{Fail NPFA} \cap \text{Officer}) = 0.25 \times 0.605 \times 0.36 = 0.05445$

2. Apply the conditional probability formula.

$P(\text{Fail NPFA} \mid \text{Officer}) = \frac{P(\text{Fail NPFA} \cap \text{Officer})}{P(\text{Officer})}$
$= \frac{0.05445}{0.60}$
$= 0.09075$

1. Calculate the result for part (ii) as a prerequisite (probability that an Officer failed the NPFA Test).

First, find the probability that a serviceman passed the NPFA Test and became an Officer:
$P(\text{Pass NPFA} \cap \text{Officer}) = 0.75 \times [0.20(0.91) + 0.63(0.72) + 0.17(0.54)] = 0.54555$
Since $P(\text{Officer}) = 0.60$ (given in part i), the probability for part (ii) is:
$P(\text{Fail NPFA} \mid \text{Officer}) = \frac{0.60 - 0.54555}{0.60} = 0.09075$

2. Use the complement rule to answer part (iii).

Since an Officer must have either passed or failed the NPFA Test, we subtract the result of part (ii) from 1:
$P(\text{Pass NPFA} \mid \text{Officer}) = 1 - P(\text{Fail NPFA} \mid \text{Officer})$
$= 1 - 0.09075 = 0.90925$

1. Calculate the unknown probability $a$ from the tree diagram (as implicitly required from part (i)).

Given $P(\text{Officer}) = 0.60$.
$P(\text{Officer}) = P(\text{Pass NPFA} \cap \text{Officer}) + P(\text{Fail NPFA} \cap \text{Officer})$
$= 0.75(0.20 \times 0.91 + 0.63 \times 0.72 + 0.17 \times 0.54) + 0.25(a \times 0.36)$
$= 0.75(0.182 + 0.4536 + 0.0918) + 0.09a$
$= 0.54555 + 0.09a$
$0.54555 + 0.09a = 0.60 \implies 0.09a = 0.05445 \implies a = 0.605$

2. Find the total probability that a randomly chosen serviceman is a Non-officer.

$P(\text{Non-officer}) = 1 - P(\text{Officer})$
$= 1 - 0.60 = 0.40$

3. Calculate the probability that a serviceman failed the NPFA test AND is a Non-officer.

$P(\text{Fail NPFA} \cap \text{Non-officer}) = P(\text{Fail NPFA}) \times P(\text{Non-officer} \mid \text{Fail NPFA})$
$= 0.25 \times [a \times 0.64 + (1 - a) \times 1.0]$
$= 0.25 \times [0.605 \times 0.64 + (1 - 0.605) \times 1.0]$
$= 0.25 \times [0.3872 + 0.395]$
$= 0.25 \times 0.7822 = 0.19555$

4. Calculate the conditional probability that a Non-officer had failed his NPFA test.

$P(\text{Fail NPFA} \mid \text{Non-officer}) = \frac{P(\text{Fail NPFA} \cap \text{Non-officer})}{P(\text{Non-officer})}$
$= \frac{0.19555}{0.40}$
$= 0.488875 \approx 0.489 \text{ (to 3 s.f.)}$