A probability question that needs some clarification

Question

I have a probability question (if it is too basic, my apologies since I am learning), I can not get my head around.

Suppose I concatenate the sentences from many books and shuffle the sentences (so we do not know the book that a sentence is from) to have a "sentence database". We assume there is no duplicated sentences, and we knew the books are written by two different writers. Now, given a phrase (sequence of words), we can find this phrase at 4 sentences in our database. And we are also given that for these 4 sentences:

sentence 1 has 20% probability is written by writer 1

sentence 2 has 60% probability is written by writer 1

sentence 3 has 50% probability is written by writer 2

sentence 4 has 40% probability is written by writer 2

The question is what is the probability of the given phrase is written by writer 1 and or writer 2?

My calculation is for the probability that the phrase is written by writer 1:

$$\frac{20\% + 60\%}{20\% + 60\% + 50\% + 40\%} = 0.470588235$$

and for writer 2:

$$1 - 0.470588235 = 0.529411765$$

My intuition tells me I am doing it wrong. But i do not where is wrong, could you please point it out.

Edit:

I thought this question is similar with calculating the probability of a outcome when flipping a coin.

So if I flip a coin 4 times, with outputs H T T H, then the probability of H is 1/2. So the formula is $$\frac{event\ occurred}{total\ number\ of\ trials}$$

Edit 2:

Where do the probabilities of the sentences come from?

It is coming from a machine learning algorithm (SVM).

What do they mean?

Because the sentences are shuffled, we lost the information about which writer wrote this sentence. However, we know the (sentence) writing styles of these two writers, so we trained a SVM by using the known writing styles as features to give each sentence a probability, the probability means this sentence has x% probability is written by writer y.

As the task is getting more difficult now. We are just given a phrase to judge (I mean assign a probability to this phrase, in case you think judge might means binary decision) which writer wrote it. (You might say can you not train another SVM to get the probability, just like what you did for sentence probabilities. Assume we can not do it.)

I had modified my question, it should be "The question is what is the probability of the given phrase is written by writer 1 or writer 2?"

Answer 1

Yes, you have a mistake.

$$ P(writer1) = P(writer1 | sentence1) P(sentence1) + \ldots + P(writer1 | sentence4) P(sentence4) $$

We know each of the $P(writer1 | sentence)$ terms: $$ P(writer1|sentence1) = 0.2$$ $$ P(writer1|sentence2) = 0.6$$ $$ P(writer1|sentence3) = 1 - P(writer2|sentence3) = 0.5$$ $$ P(writer1|sentence4) = 1 - P(writer2|sentence4) = 0.6$$

Assuming the probability of each sentence being the one the phrase is in is the same (I think this is fair assumption) $\implies P(sentence j) = \frac{1}{4}$

Then, $$P(writer1) = (0.2 + 0.6 + 0.5 + 0.6) \cdot \frac{1}{4} = 0.475$$

Note that its just a coincidence that you happened to be close to the right answer. I suppose you could think of this in the way that you suggest too:

$$ \frac{events}{trials}$$

But in that case:

$$ events = .2 + .6 + .5 + .4 $$ (the fraction of the time the event that writer 1 is the author)

$$ trials = 1 + 1 + 1 + 1$$

You will get the same answer, but I think the conditional probability way is a bit easier to think about.