In Sunny Island Republic, all male citizens are required to undergo a National Physical Fitness Assessment (NPFA) Test before enlisting for National Service in the army. Based on historical data, 75% of males who attempt the NPFA Test will pass the test and they are awarded either a Bronze, Silver or Gold standard. The performance at the NPFA Test is known to influence the chances of these National Servicemen being promoted to the rank of an Officer. For male citizens who fail the NPFA Test, they are required to undergo a Physical Assessment Test (PAT) at the end of their basic training. It is known that those who pass the PAT will have a chance to be promoted to the rank of an Officer. The information is summarized in the tree diagram below.

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.

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