For an n-Gram model with n>2, do we need more context at end of each sentence?

Jurafsky's book says we need to add context to left and right of a sentence:

Does this mean,

for example, if we've a corpus of three sentences: "John read Moby Dick", "Mary read a different book", and "She read a book by Cher"; and after training our tri-gram model on this corpus of three sentences, we need to evaluate the probability of a sentence "John read a book", i.e. to find $$P(John\; read\; a\; book)$$ as below,

$$P(John\; read\; a\; book)$$
$$=P(John\; read\; a\; book<\backslash s><\backslash s>)$$
$$=P(John|) P(read|John) \; P(a|John\; read) P(book|read\; a) P(<\backslash s>|a\; book)\; P(<\backslash s>|book<\backslash s>)$$
$$=\frac{1}{3}\frac{1}{1}\frac{1}{1}\frac{1}{2}\frac{0}{1}\frac{1}{1}$$ (without smoothing)

It would be great, if you let me know if the above understanding is correct?