Probability Problem Set

Instructions

You need to include these 3 parts in each of your responses to the problem sets.

1. Set Up theEquations
2. Use the Equations/Mathematical Manipulation and Operations
3. Discuss the Results

Problem Set

1. Probabilities of sequences.

Assume that the four bases A, C, T, and G occur with equal likelihood in a DNA sequence of nine monomers.

1. What is the probability of finding the sequence AAATCGAGT through random chance?
2. What is the probability of finding the sequence AAAAAAAAA through random chance?
3. What is the probability of finding any sequence that has four A's, two T's, two G's, and one C, such as that in (a)?

2. The probability of a sequence (given a composition).

A scientist has constructed a secret peptide to carry a message. You know only the composi-tion of the peptide, which is six amino acids long. It contains one serine S, one threonine T, one cysteine C, one arginine R, and two glutamates E. What is the probability that the sequence SECRET will occur by chance?

3. Computing a mean and variance.

Consider the probability distribution p(x) = axⁿ, 0 x 1, for a positive integer n.

1. Derive an expression for the constant a, to normalize p(x).
2. Compute the average xas a function of n.
3. Compute ²= x² x²as a function of n.

4. The probabilities of identical sequences of amino acids.

You are comparing protein amino acid sequences for homology. You have a twenty-letter alphabet (twenty different amino acids). Each sequence is a string n letters in length. You have one test sequence and s different database sequences. You may find any one of the twenty different amino acids at any position in the sequence, independent of what you find at any other position. Let p represent the probability that there will be a 'match' at a given position in the two sequences.

1. In terms of s, p, and n, how many of the s sequences will be perfect matches (identical residues at every position)?
2. How many of the s comparisons (of the test sequence against each database sequence) will have exactly one mismatch at any position in the sequences?

5. The distribution of scores on dice.

Suppose that you have n dice, each a different color, all unbiased and six-sided.

1. If you roll them all at once, how many distinguishable outcomes are there?
2. Given two distinguishable dice, what is the most probable sum of their face values on a given throw of the pair? (That is, which sum between two and twelve has the greatest number of different ways of occurring?)
3. What is the probability of the most probable sum?

6. The Maxwell-Boltzmann probability distribution function.

According to the kinetic theory of gases, the energies of molecules moving along the x-direction are given by x = (1/2)mv_x², where m = mass and v_x is the velocity in the x-direction. The distribution of particles over velocities is given by the Boltzmann law, p(v_x) = e^mv_x^{2 /2kT}. This is the Maxwell-Boltzmann distribution (velocities may range from to +).

1. Write the probability distribution p(v_x), so that the Maxwell-Boltzmann distribution is correctly normalized.
2. Compute the average energy (1/2)mv_x².
3. What is the average velocity v_x?

What is the average momentum mv_x?