An author writes a good book with a probability of \[\dfrac{1}{2}\]. If it is good it is published with a probability of \[\dfrac{3}{2}\]. If it is not, it is published with a probability of \[\dfrac{1}{4}\]. Find the probability that he will get at least one book published if he writes two.

Hint: As we know that the probability means simply how likely something is to happen. When we are not sure about the outcome of an event, we can talk about the probabilities of certain outcomes.

The probability of an event is a number between \[0\] & \[1\], where \[0\] indicates the impossibility of the event and \[1\] indicates certainty.
If you know the probability of an event occurring, it is easy to compute the probability that the event does not occur. If P(A) is the probability of Event A, then \[(1 - P(A))\] is the probability that the event does not occur.

Probability of an event is given by: -
Therefore \[\Pr obability = \left( {\dfrac{{Possible\,outcomes}}{{Total\,outcomes}}} \right)\]

Complete step by step solution:
Given,
Probability of a good book \[ = \dfrac{1}{2}\]
Probability of a good book being published \[ = \dfrac{1}{4}\]

Let us find probability for one book to be published \[ = \] (Probability of good book and published \[ + \]Probability of a book i.e. not good and published).

\[\dfrac{1}{2} \times \dfrac{2}{3} + \dfrac{1}{2} \times \dfrac{1}{4}\]
\[ = \dfrac{1}{3} + \dfrac{1}{8}\]
\[ = \dfrac{{8 + 3}}{{24}} = \dfrac{{11}}{{24}}\]

Probability no book published \[ = \] \[1 - \]Probability of book being published.
\[ = 1 - \dfrac{{11}}{{24}}\]
\[ = \,\,\dfrac{{24 - 11}}{{24}}\,\, = \,\,\dfrac{{13}}{{24}}\]

Probability that two books are not published \[ = \] \[1 - \] Probability that two books are not published
\[ = \,1\, - \,\dfrac{{169}}{{576}}\]
\[ = \,\dfrac{{576 - 169}}{{576}}\, = \,\dfrac{{407}}{{576}}\]

Note: We use multiplication in the probability until the case is not complete, we use \[' + '\] sign for the ‘either, or’ case i.e. when a particular case or probability is complete and we are considering other possibilities.