A 50.0 mL sample of a solution of a monoprotic acid is titrated with a 0.115 M NaOH solution. The titration curve above was obtained. The concentration of the monoprotic acid is about ________ mol/L. SHOW YOUR WORK

Derive the conditional distribution of Y given X = 1 and compute its value at Y = 3. That is, compute . Show the formula you derived for  on your scratch paper.

Compute the expected time (hours) until both Adam and Ben complete their services from when Cindy’s service begins.

Compute the probability that the gambler is ruined (loses all money). Provide three digits after the decimal point. Show your work in answering on your scratch paper.

Consider a DTMC with state space {A, B, C, D, E, F, G} and the following state transitions: A

What is the expected time in hours until the first service completion after Cindy’s service begins?

Identify all communicating classes. For instance, for a DTMC with states {0,1,2}, if 0 and 1 communicate with each other, but neither does with 2, then the states can be classified in communicating classes as {0,1}, {2}.

Suppose you found two eggs from the coop today. Compute the expected number of eggs to find tomorrow. Show your work on the scratch paper.

Identify all recurrent states.

A gambler starts with $4 and plays a coin-flipping game, where: On each bet, the gambler wins $1 with probability 0.4 and loses $1 with probability 0.6. The game continues until the gambler either reaches $8 or loses all money ($0). Let us define a DTMC whose state corresponds to the gambler’s capital.