# Set of 5 coin tosses giving specific set of data . And how to maximize probability on that data?

The set of data is $S={1,1,0,1,1}$

What is the probability this set is going to be accomplished, giving that $P(x=1)=0.5$ ?

My guess is that it's $\frac {1}{2^5}$

Normally, the formula I guess would be

$$P(S) = P(x_1=1) * P(x_2=1|x_1=1) * P(x_3=0|x_1=1,x_2=1) ...$$

But since events are conditional independent

$$P(x_2=1|x_1=1) = \frac {P(x_2=1) * P(x_1=1)}{P(x_1=1)} = P(x_2=1)$$

am I right or terribly wrong ?

How to maximize probability of given set ? What should the value of $P(x=1)$ should be for $P(S)=max$ ( probability of given set to be max )

You should maximize the value $$W(p)=p^4(1-p).\tag{1}$$ The global maximum must be either a local maximum in the interior of the domain $[0,1]$, or lie on the boundary of the domain.
In the case of function (1) it is the local maximum, which can be found by solving the equation: $$\frac{dW}{dp}=4p^3-5p^4=p^3(4-5p)=0,$$ resulting in the value $p_{\rm max}=\frac{4}{5}$, which was also intuitively clear.