While reading a table of cumulative binomial probabilities, if I need to find the probability of, for example, exactly 4 successful events happening and all the rest failures occurring, how would I go about that ?
[Math] Reading Binomial Tables
binomial distributionprobability
Related Solutions
Let us use a Poisson model, since both of you are doing so, with $\lambda$ somewhere near $\dfrac{110}{52}$.
The negative binomial approach: Call a week bad if the week has at least one tornado. We can ask: what is the probability that the fourth week is the third bad week? The negaive binomial, with $p=1-e^{-\lambda}$, answers this question. But I do not think that is the ordinary english meaning of the question as stated.
I would suggest the following. The number of tornadoes in the first three weeks is Poisson parameter $3\lambda$, where $\lambda$ is as above. Then the event "third tornado in the fourth week" can happen in $3$ ways:
(i) $0$ tornadoes in first three weeks, at least $3$ in fourth week.
(ii) $1$ tornado in first three weeks, at least $2$ in fourth.
(iii) $2$ tornadoes in first three weeks, at least $1$ in the fourth.
Add up.
The calculation is not too bad. The probability of $k$ tornadoes in the first three weeks is $e^{-3\lambda}\dfrac{(3\lambda)^k}{k!}$.
For the probability of the "at least" part, do the usual thing. For example, the probability of at least $1$ in the fourth is $1-e^{-\lambda}$.
Remark: Using a Poisson model is not exactly reasonable! Tornadoes come in bunches. And I do not know about Texas, but in Canada there are very few tornadoes in January.
Hint: Add up the area you want to calculate! 
Best Answer
The table you show is for when $n=20$ and is cumulative. If you are interested in finding the probability of at most $4$ successful events and the rest failures... you look at the row which begins with the bold 4, then look across until reaching the column which begins with the bold 0.xx where 0.xx is the probability of success. Trace your finger across the row and down the column to see where they meet. For example, at most $4$ successes out of $20$ where probability of success is $0.25$ occurs with probability approximately $0.415$.
Since you are interested in the value for exactly $4$ successes rather than at most $4$ successes, you can subtract the value right above it. I.e. $Pr(X= 4) = Pr(X\leq 4) - Pr(X\leq 3)$, which for the case where $p=0.25$ yields approximately $0.415 - 0.225 = 0.19$.
Doing this directly yields $\binom{20}{4}0.25^40.75^{16} \approx 0.189685\cdots$ confirming the result above.
If you were interested in probabilities where $n$ were some number different than $20$ then you will need a different table. If you are in a scenario where you have a calculator or calculator software available, I see no reason to bother with tables in the first place and would instead use the usual formulas directly.