A coin will be flipped repeatedly until the sequence TTH (tail/tail/head) comes up. Successive flips are independent, and the coin has probability p of coming up heads. Let N, TTH be the number of coin flips until TTH first appears. What value of p minimizes Ex[N, TTH]

Step-by-step explanation:

The calculation of the value of p minimizes is shown below:-

We will assume the probability of coming heads be p

p(H) = p

p(T) = 1 - P

Now, H and T are only outcomes of flipping a coin

So,

P(TTH) = (1 - P) = (1 - P) (1 - P) P

= (1 + P^2 - 2 P) P

= P^3 - 2P^2 + P

In order to less N,TTH

we need to increase P(TTH)

The equation will be

3P^2 - 4P + 1 = 0

(3P - 1) (P - 1) = 0

P = 1 and 1 ÷ 3

For P(TTH) to be maximum

= 6P - 4

and

(6P - 4) is negative which is for

answer: 2 litters juice bottle

half of a gallon is 50% of a full gallon 2 litters of juice is 52% of a full gallon

hope this ^_^

triangle abc is reflected on the line x=-6.5 then reflected again on the x axis

step-by-step explanation: