A biased coin gives a head probability P the expected number of times does one need to make to get the pattern of HTH is 36.25 what is the value of P rounded to the nearest 0.01.

Answer: P = 0.2

Step-by-step explanation:

let us define the expectations as;

Eнтн = no of flips to obtain HTH

Eн, тн = no of flips to obtain HTH where H is flipped

Eнт, н = no of flips to obtain HTH where HT is flipped

so let P and q represent the success and failure of probabilities

this gives;

Eнтн = 2 + P²Eн, тн + PqEнт, н + PqEн, тн + q²Eнтн

Eн, тн = 1 + PEн, тн + qEнт, н

Eн, тн = (1 + qEнт, н) / q

Eнт, н = 1 + p*0 + qEнтн = 1 + qEнтн

from this expression we have that;

Eнтн = (2 + (P² + Pq) (1/q + 1) + Pq) / (1-qP² + 2Pq² + q²)

E (x) = 1/P + 1/P²q

= 36.25

therefore the probability is P = 0.2

P = 0.2