An electric instrument consists of two units. Each unit must function independently for the instrument to operate. The probability that the first unit functions is 0.9 and that of the second unit is 0.8. The instrument is switched on and it fails to operate. If the probability that only the first unit failed and second unit is functioning is $$p$$, then $$98p$$ is equal to _________.

Let us denote the events as follows: $$A$$ is the event that the first unit works and $$B$$ is the event that the second unit works. We are told that the two units behave independently, so their probabilities multiply when we need the intersection of their events.

We have the probabilities of each unit functioning:

$$P(A)=0.9,\qquad P(B)=0.8.$$

If a unit does not work, we denote its failure with a prime. Thus

$$P(A')=1-P(A)=1-0.9=0.1,$$

$$P(B')=1-P(B)=1-0.8=0.2.$$

The instrument fails to operate whenever at least one of the units fails. The failure event for the whole instrument is therefore the union $$A'\cup B'$$. By the Addition Law of Probability, for independent events, we can compute

$$P(A'\cup B')=P(A')+P(B')-P(A'B').$$

But independence gives $$P(A'B')=P(A')P(B')=0.1\times0.2=0.02,$$ so

$$P(A'\cup B')=0.1+0.2-0.02=0.28.$$

There is a simpler equivalent route: the instrument works only when both units work, i.e. on $$AB$$. Hence

$$P(A'\cup B')=1-P(AB)=1-P(A)P(B)=1-0.9\times0.8=1-0.72=0.28,$$

which matches the previous calculation.

We want the conditional probability that only the first unit failed while the second unit still works, given that the instrument has failed. Symbolically, this is

$$p=P(A'B\mid A'\cup B').$$

By definition of conditional probability,

$$P(A'B\mid A'\cup B')=\frac{P(A'B)}{P(A'\cup B')}.$$

Again using independence,

$$P(A'B)=P(A')P(B)=0.1\times0.8=0.08.$$

Substituting the values we have just obtained,

$$p=\frac{0.08}{0.28}=\frac{8}{28}=\frac{2}{7}.$$

The question asks for the value of $$98p$$, so we multiply:

$$98p=98\times\frac{2}{7}=14\times2=28.$$

Hence, the correct answer is Option 28.