Question

There are three questions, and three choices of answer (A, B and C) for each question. If only one of the possible choices (A, B or C) is correct for each question, what is the probability of getting one answer correct?

The answer to this question is 4/9 but I don't understand it

Answer 1

Please see below.

Explanation:

(1) There are `3^3=27` possibilities for total chioces.
(2) Suppose the three question is marked as (I),(II) and (III).
If the answer for question (I) is correct, those for (II) and (III) are wrong.

Since (II) and (III) have `color(red)"independent"` wrong answers, ,the answering pattern is `1*2*2=4`.
This applies when (II) or (III) is collect, so there are `4*3=12` possibilities to get one correct answer.

(3) In conclusion, the probabilitiy is `12/27=4/9`.

[What do you mean by "independent"?]
Two events `A` and `B` are `color(red) "independent"` if and only if `color(red)(P(AnnB)=P(A)P(B))`, where `P(X)` is the possibility `X` happens. (This is the definition of independence.)
Thus, the number of events `n(X)` follows `color(blue)(n(AnnB)=n(A)n(B))`
if and only if `A` and `B` are independent.

For example, if you roll two dices `A` and `B`, the possibilities to have a 1 is
`P(A)=1/6`, `P(B)=1/6` and `P(AnnB)`=`1/6xx1/6=1/36`
and they are independent.