Probability

Problem set: 1

1. Your PH5080 (Statistical Physics) class has 65 students. What is the probability that at leasttwo of your classmates have the same birthday?

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2. Assume that the four bases A, C, T, and G occur with equal likelihood in a DNA sequenceof nine monomers. What is the probability of(a) finding the sequence AAATCGAGT through random chance?(b) finding the sequence AAAAAAAAA through random chance?(c) finding any sequence that has four As, two Ts, two Gs, and one C, such as in (a)?

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3. A biochemist has constructed a secret peptide to carry a message. You know only the com-position of the peptide, which is six amino acids long. It contains one serine S, one threonineT, one cysteine C, one arginine R, and two glutamates E. What is the probability that thesequence SECRET will occur by chance?

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4. How many times a fair sided die needs to be rolled so that that the probability is 2/3 that atleast one 2 will appear?

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5. Suppose you roll a die three times. What is the probability of(a) getting a total of two 5s from all three rolls of the dice?(b) getting a total of at least two 5s from all three rolls of the die?

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6. If you flip an unbiased green coin and an unbiased red coin five times each, what is theprobability of getting four red heads and two green tails?

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7. In a game show there are three closed doors: one hides a car, and two hide goats. You pointto a door (say C). The show host, knowing what’s behind each door, now opens either doorA or B, to show you a goat; say it’s door A. To win a car, you now get a final choice: shouldyou stick with your original choice C, or should you now switch and choose door B?

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8. Calculate the characteristic function, mean, and variance of the following probability densityfunctions (PDF):(a) Uniform p(x) = 1

2afor −a < x < a and p(x) = 0 otherwise. (b) Exponential

p(x) = exp(−ax) for x ≥ 0. Is this a PDF?(c) Gaussian p(x) = a exp(−bx2)(d) Laplace p(x) = 1

2aexp(−|x|/a).

(e) Cauchy/Lorentz p(x) = (a/π)/(x2 + a2).(f) Rayleigh p(x) = x

a2exp(− x2

2a2) for x ≥ 0.

(g) Maxwell p(x) =√

2πx2

a3exp(− x2

2a2) for x ≥ 0.

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