Let O=TTHHTHTTHT

and knowing p=P[head]

$ P[O;p] = p^4(1-p)^6 $

$ P_{ML}=max(p^4(1-p)^6) $

To find the max of a function one could find the derivative of the function and set the it zero and then solve for the variable that was used in finding the derivative.

$ \frac{d}{dp}(p^4(1-p)^6) = 2p^3(p-1)^5(5p - 2) $

$ 2*p^3(p-1)^5(5p - 2) = 0, p = 2/5 $

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Basic linear algebra uncovers and clarifies very important geometry and algebra.

Dr. Paul Garrett