I had another inquiry, for the table of probabilities that I presented I saw another trend that some of the probabilities are the same, and that the pattern continues down. I don't know why this happens. Could someone explain why this happens?? I have attached the probabilities table below highlighting the probabilities that are the same. Thanks for your help!
That looks rigged by the consequence of \(p=\frac14\) in your binomial distribution.
\[ \text{Observe from direct computation that }3\times \binom71 = \binom72. \]
\[ \text{Hence} \] \begin{align*}
\binom72 (0.25)^2 (0.75)^5 &= 3\times \binom71 \times (0.25)^2(0.75)^5\\
&= (3\times 0.25) \times \binom71 (0.25)(0.75)^5\\
&= 0.75 \times \binom71 (0.25) (0.75)^5\\
&= \binom71 (0.25)(0.75)^6.
\end{align*}
Similar algebraic tricks can be done to prove the others. (It may be worth remarking that \( \binom72 = \frac34\times \binom82 \).) But basically, the trick is just shuffling the terms around.