Chapter 1, Principles of Probability Video Solutions, Molecular Driving Forces

Problem 1

Combining independent probabilities. You have applied to three schools: University of California at San Francisco (UCSF), Duluth School of Mines (DSM), and Harvard (H). You guess that the probabilities you'll be accepted are $p(\mathrm{UCSF})=0.10, p(\mathrm{DSM})=0.30$, and $p(\mathrm{H})=0.50$. Assume that the acceptance events are independent.
(a) What is the probability that you get in somewhere (at least one acceptance)?
(b) What is the probability that you will be accepted by both Harvard and Duluth?

Rashmi Sinha

Numerade Educator

00:35

Problem 2

Probabilities of sequences. Assume that the four bases A, C, T, and G occur with equal likelihood in a DNA sequence of nine monomers.
(a) What is the probability of finding the sequence AAATCGAGT through random chance?
(b) What is the probability of finding the sequence AAAAAAAAA through random chance?
(c) 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)?

Jeff Vermeire

Numerade Educator

02:43

Problem 3

The probability of a sequence (given a composition). A scientist has constructed a secret peptide to carry a message. You know only the composition of the peptide, which is six amino acids long. It contains one serine $\mathbf{S}$, one threonine $\mathbf{T}$, one cysteine $\mathbf{C}$, one arginine $\mathbf{R}$, and two glutamates $\mathbf{E}$. What is the probability that the sequence SECRET will occur by chance?

Susan Hallstrom

Numerade Educator

02:22

Problem 4

Combining independent probabilities. You have a fair six-sided die. You want to roll it enough times to ensure that a 2 occurs at least once. What number of rolls $k$ is required to ensure that the probability is at least $2 / 3$ that at least one 2 will appear?

Sanchit Jain

Numerade Educator

01:17

Problem 5

Predicting compositions of independent events. Suppose you roll a fair six-sided die three times.
(a) What is the probability of getting a 5 twice from all three rolls of the dice?
(b) What is the probability of getting a total of at least two 5 's from all three rolls of the die?

Massimo Antonelli

Numerade Educator

04:43

Problem 6

Computing a mean and variance. Consider the probability distribution $p(x)=a x^{n}, 0 \leq x \leq 1$, for a positive integer $n$.
(a) Derive an expression for the constant $a$, to normalize $p(x)$.
(b) Compute the average $\langle x\rangle$ as a function of $n$.
(c) Compute $\sigma^{2}=\left\langle x^{2}\right\rangle-\langle x\rangle^{2}$ as a function of $n$.

Ryan Finn

Numerade Educator

02:23

Problem 7

Computing the average of a probability distribution. Compute the average $\langle i\rangle$ for the probability distribution function shown in Figure $1.17$.

Emily Bender

Numerade Educator

04:40

Problem 8

Predicting coincidence. Your statistical mechanics class has 25 students. What is the probability that at least two classmates have the same birthday?

Oluwadamilola Ameobi

Numerade Educator

00:58

Problem 9

The distribution of scores on dice. Suppose that you have $n$ dice, each a different color, all unbiased and sixsided.
(a) If you roll them all at once, how many distinguishable outcomes are there?
(b) 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 2 and 12 has the greatest number of different ways of occurring?)
(c) What is the probability of the most probable sum?

Hossam Mohamed

Numerade Educator

01:59

Problem 10

The probabilities of identical sequences of amino acids. You are comparing protein amino acid sequences for homology. You have a 20-letter alphabet (20 different amino acids). Each sequence is a string $n$ letters in length. You have one test sequence and $s$ different data base sequences. You may find any one of the 20 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.
(a) In terms of $s, p$, and $n$, how many of the $s$ sequences will be perfect matches (identical residues at every position)?
(b) 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?

Sana Riaz

Numerade Educator

03:46

Problem 11

The combinatorics of disulfide bond formation. A protein may contain several cysteines, which may pair together to form disulfide bonds as shown in Figure 1.18. If there is an even number $n$ of cysteines, $n / 2$ disulfide bonds can form. How many different disulfide pairing arrangements are possible?

Sana Riaz

Numerade Educator

02:02

Problem 12

Predicting combinations of independent events. If you flip an unbiased green coin and an unbiased red coin five times each, what is the probability of getting four red heads and two green tails?

Fan Yang

Numerade Educator

01:10

Problem 13

A pair of aces. What is the probability of drawing two aces in two random draws without replacement from a full deck of cards?

Raj Bala

Numerade Educator

02:37

Problem 14

Average of a linear function. What is the average value of $x$, given a distribution function $q(x)=c x$, where $x$ ranges from zero to one, and $q(x)$ is normalized?

Bahar Tehranipoor

Numerade Educator

01:40

Problem 15

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 $\varepsilon_{x}=(1 / 2) m v_{x}^{2}$, where $m$ is mass and $v_{x}$ is the velocity in the $x$ direction. The distribution of particles over velocities is given by the Boltzmann law, $p\left(v_{x}\right)=e^{-m v_{x}^{2} / 2 k T}$. This is the Maxwell-Boltzmann distribution (velocities may range from $-\infty$ to $+\infty$ ).
(a) Write the probability distribution $p\left(v_{x}\right)$, so that the Maxwell-Boltzmann distribution is correctly normalized.
(b) Compute the average energy $\left\langle\frac{1}{2} m v_{x}^{2}\right\rangle$.
(c) What is the average velocity $\left\langle v_{x}\right\rangle$ ?
(d) What is the average momentum $\left\langle m v_{x}\right\rangle$ ?

John Nicolle

Numerade Educator

01:59

Problem 16

Predicting the rate of mutation based on the Poisson probability distribution function. The evolutionary process of amino acid substitution in proteins is sometimes described by the Poisson probability distribution function. The probability $p_{s}(t)$ that exactly $s$ substitutions at a given amino acid position occur over an evolutionary time $t$ is
$$
p_{s}(t)=\frac{e^{-\lambda t}(\lambda t)^{s}}{s !},
$$
where $\lambda$ is the rate of amino acid substitution per site per unit time. Fibrinopeptides evolve rapidly: $\lambda_{F}=9.0$ substitutions per site per $10^{9}$ years. Lysozyme is intermediate: $\lambda_{L} \approx 1.0$. Histones evolve slowly: $\lambda_{H}=0.010$ substitutions per site per $10^{9}$ years.
(a) What is the probability that a fibrinopeptide has no mutations at a given site in $t=1$ billion years?
(b) What is the probability that lysozyme has three mutations per site in 100 million years?
(c) We want to determine the expected number of mutations $\langle s\rangle$ that will occur in time $t$. We will do this in two steps. First, using the fact that probabilities must sum to one, write $\alpha=\sum_{s=0}^{\infty}(\lambda t)^{s} / s !$ in a simpler form.
(d) Now write an expression for $\langle s\rangle$. Note that
$$
\sum_{s=0}^{\infty} \frac{s(\lambda t)^{s}}{s !}=(\lambda t) \sum_{s=1}^{\infty} \frac{(\lambda t)^{s-1}}{(s-1) !}=\lambda t \alpha
$$
(e) Using your answer to part (d), determine the ratio of the expected number of mutations in a fibrinopeptide to the expected number of mutations in histone protein, $\langle s\rangle_{\mathrm{fib}} /\langle s\rangle_{\mathrm{his}}[6]$.

Sana Riaz

Numerade Educator

01:53

Problem 17

Probability in court. In forensic science, DNA fragments found at the scene of a crime can be compared with DNA fragments from a suspected criminal to determine the probability that a match occurs by chance. Suppose that DNA fragment $A$ is found in $1 \%$ of the population, fragment $B$ is found in $4 \%$ of the population, and fragment $C$ is found in $2.5 \%$ of the population.
(a) If the three fragments contain independent information, what is the probability that a suspect's DNA will match all three of these fragment characteristics by chance?
(b) Some people believe such a fragment analysis is flawed because different DNA fragments do not represent independent properties. As before, suppose that fragment $A$ occurs in $1 \%$ of the population. But now suppose the conditional probability of $B$, given $A$, is $p(B \mid A)=0.40$ rather than $0.040$, and $p(C \mid A)=0.25$ rather than $0.025$. There is no additional information about any relationship between $B$ and $C$. What is the probability of a match now?

Lucas Finney

Numerade Educator

03:35

Problem 18

Flat distribution. Given a flat distribution, from $x=$ $-a$ to $x=a$, with probability distribution $p(x)=1 /(2 a)$ :
(a) Compute $\langle x\rangle$.
(b) Compute $\left\langle x^{2}\right\rangle$.
(c) Compute $\left\langle x^{3}\right\rangle$.
(d) Compute $\left\langle x^{4}\right\rangle$.

Amany Waheeb

Numerade Educator

01:02

Problem 19

Family probabilities. Given that there are three children in a family, what is the probability that:
(a) two are boys and one is a girl?
(b) all three are girls?

Raj Bala

Numerade Educator

01:53

Problem 20

Evolutionary fitness. Suppose that the probability of having the dominant allele (D) in a gene is $p$ and the probability of the recessive allele $(\mathbf{R})$ is $q=1-p$. You have two alleles, one from each parent.
(a) Write the probabilities of all the possibilities: DD, DR, and RR.
(b) If the fitness of $\mathrm{DD}$ is $f_{\mathrm{DD}}$, the fitness of $\mathrm{DR}$ is $f_{\mathrm{DR}}$, and the fitness of $R R$ is $f_{R R}$, write the average fitness in terms of $p$.

John Barone

Numerade Educator

03:07

Problem 21

Ion-channel events. A biological membrane contains $N$ ion-channel proteins. The fraction of time that any one protein is open to allow ions to flow through is q. Express the probability $P(m, N)$ that $m$ of the channels will be open at any given time.

Jessica Wooten

Numerade Educator

04:57

Problem 22

Joint probabilities: balls in a barrel. For Example $1.10$, two green balls and one red ball drawn from a barrel without replacement:
(a) Compute the probability $p(R G)$ of drawing one red and one green ball in either order.
(b) Compute the probability $p(G G)$ of drawing two green balls.

Amany Waheeb

Numerade Educator

03:30

Problem 23

Sports and weather. The San Francisco football team plays better in fair weather. They have a $70 \%$ chance of winning in good weather, but only a $20 \%$ chance of winning in bad weather.
(a) If they play in the Super Bowl in Wisconsin and the weatherman predicts a $60 \%$ chance of snow that day, what is the probability that San Francisco will win?
(b) Given that San Francisco lost, what is the probability that the weather was bad?

Ankit Gupta

Numerade Educator

02:47

Problem 24

Monty Hall's dilemma: a game show problem. You are a contestant on a game show. There are three closed doors: one hides a car and two hide goats. You point to one door, call it $C$. The gameshow host, knowing what's behind each door, now opens either door $A$ or $B$, to show you a goat; say it's door $A$. To win a car, you now get to make your final choice: should you stick with your original choice $C$, or should you now switch and choose door B? (New York Times, July 21, 1991; Scientific American, August 1998.)

James Kiss

Numerade Educator

03:14

Problem 25

Probabilities of picking cards and rolling dice.
(a) What is the probability of drawing either a queen or a heart in a normal deck of 52 cards?
(b) What is the probability $P$ of getting three 7 's and two 4 's on five independent rolls of a die?

Chai Santi

Numerade Educator

03:23

Problem 26

Probability and translation-start codons. In prokaryotes, translation of mRNA messages into proteins is most often initiated at start codons on the mRNA having the sequence AUG. Assume that the mRNA is single-stranded and consists of a sequence of bases, each described by a single letter A, C, U, or G.
Consider the set of all random pieces of bacterial mRNA of length six bases.
(a) What is the probability of having either no A's or no U's in the mRNA sequence of six base pairs long?
(b) What is the probability of a random piece of mRNA having exactly one $\mathbf{A}$, one $\mathbf{U}$, and one $\mathbf{G}$ ?
(c) What is the probability of a random piece of mRNA of length six base pairs having an A directly followed by a U directly followed by a G; in other words, having an AUG in the sequence?
(d) What is the total number of random pieces of mRNA of length six base pairs that have exactly one $\mathbf{A}$, exactly one $\mathbf{U}$, and exactly one $\mathbf{G}$, with $\mathbf{A}$ appearing first, then the $\mathbf{U}$, then the $\mathbf{G}$ ? (e.g., AXXUXG)

Sana Riaz

Numerade Educator

07:22

Problem 27

DNA synthesis. Suppose that upon synthesizing a molecule of DNA, you introduce a wrong base pair, on average, every 1000 base pairs. Suppose you synthesize a DNA molecule that is 1000 bases long.
(a) Calculate and draw a bar graph indicating the yield (probability) of each product DNA, containing 0,1 , 2 , and 3 mutations (wrong base pairs).
(b) Calculate how many combinations of DNA sequences of 1000 base pairs contain exactly 2 mutant base pairs.
(c) What is the probability of having specifically the 500 th base pair and the 888 th base pair mutated in the pool of DNA that has only two mutations?
(d) What is the probability of having two mutations side-by-side in the pool of DNA that has only two mutations?

Anna Golebiewski

Numerade Educator

06:54

Problem 28

Presidential election. Two candidates are running for president. Candidate $A$ has already received $80 \mathrm{elec}-$ toral votes and only needs 35 more to win. Candidate $B$ already has 50 votes, and needs 65 more to win.
Five states remain to be counted. Winning a state gives a candidate 20 votes; losing gives the candidate zero votes. Assume both candidates otherwise have equal chances to win in those five states.
(a) Write an expression for $W_{A}$, total, the number of ways A can succeed at winning 40 more electoral votes.
(b) Write the corresponding expression for $W_{B, \text { total. }}$.
(c) What is the probability candidate $A$ beats candidate $B$ ?

Ivan Kochetkov

Numerade Educator