 1.1: Consider rolling a sixsided die. Let A be the set of outcomes wher...
 1.2: Let A and B be two sets. (a) Show that (b) Show that ( A n B) C = A...
 1.3: Prove the identity Solution. If x belongs to the set on the left, t...
 1.4: Cantor's diagonalization argument. Show that the unit interval [0, ...
 1.5: Out of the students in a class, 60% are geniuses, 70% love chocolat...
 1.6: A sixsided die is loaded in a way that each even face is twice as ...
 1.7: A foursided die is rolled repeatedly, until the first time (if eve...
 1.8: You enter a special kind of chess tournament, in which you play one...
 1.9: A partition of the sample space n is a collection of disjoint event...
 1.10: Show the formula p( (A n BC) u (AC n B)) = P(A) + P(B)  2P(A n B),...
 1.11: Bonferroni's inequality. (a) Prove that for any two events A and B,...
 1.12: The inclusionexclusion formula. Show the following generalizations...
 1.13: Continuity property of probabilities. (a) Let AI , A2, ... be an in...
 1.14: We roll two fair 6sided dice. Each one of the 36 possible outcomes...
 1.15: A coin is tossed twice. Alice claims that the event of two heads is...
 1.16: We are given three coins: one has heads in both faces, the second h...
 1.17: A batch of one hundred items is inspected by testing four randomly ...
 1.18: Let A and B be events. Show that P(A n B I B) = P(A I B), assuming ...
 1.19: Alice searches for her term paper in her filing cabinet. which has ...
 1.20: How an inferior player with a superior strategy can gain an advanta...
 1.21: Two players take turns removing a ball from a jar that initially co...
 1.22: Each of k jars contains m white and n black balls. A ball is random...
 1.23: We have two jars, each initially containing an equal number of ball...
 1.24: The prisoner's dilemma. The release of two out of three prisoners h...
 1.25: A twoenvelopes puzzle. You are handed two envelopes. and you know ...
 1.26: The paradox of induction. Consider a statement whose truth is unkno...
 1.27: Alice and Bob have 2n + 1 coins, each coin with probability of head...
 1.28: Conditional version of the total probability theorem. Let Cl , . , ...
 1.29: Let A and B be events with peA) > 0 and PCB) > O. We say that an ev...
 1.30: A hunter has two hunting dogs . One day, on the trail of some anima...
 1.31: Communication through a noisy channel. a source transmits a message...
 1.32: The king's sibling. The king has only one sibling. What is the prob...
 1.33: Using a biased coin to make an unbiased decision. Alice and Bob wan...
 1.34: An electrical system consists of identical components, each of whic...
 1.35: Reliability of a koutofin system. A system consists of n identic...
 1.36: A power utility can supply electricity to a city from n different p...
 1.37: A cellular phone system services a population of n l "voice users" ...
 1.38: The problem of points. Telis and Wendy play a round of golf ( 18 ho...
 1.39: A particular class has had a history of low attendance. The annoyed...
 1.40: Consider a coin that comes up heads with probability P and tails wi...
 1.41: Consider a game show with an infinite pool of contestants, where at...
 1.42: Gambler's ruin. A gambler makes a sequence of independent bets. In ...
 1.43: Let A and B be independent events. Use the definition of independen...
 1.44: Let A. B. and C be independent events, with P( C) > O. Prove that A...
 1.45: Assume that the events AI , A2, A3, A4 are independent and that P(A...
 1.46: Laplace's rule of succession. Consider m + 1 boxes with the kth box...
 1.47: Binomial coefficient formula and the Pascal triangle. (a) Use the d...
 1.48: Consider an infinite sequence of trials. The probability of success...
 1.49: De Mere's puzzle. A sixsided die is rolled three times independent...
 1.50: The birthday problem. Consider n people who are attending a party. ...
 1.51: An urn contains m red and n white balls. (a) We draw two balls rand...
 1.52: We deal from a wellshuffled 52card deck. Calculate the probabilit...
 1.53: Ninety students, including Joe and Jane, are to be split into three...
 1.54: Twenty distinct cars park in the same parking lot every day. Ten of...
 1.55: Eight rooks are placed in distinct squares of an 8 x 8 chessboard, ...
 1.56: An academic department offers 8 lower level courses: {L1 , L2, .......
 1.57: How many 6word sentences can be made using each of the 26 letters ...
 1.58: We draw the top 7 cards from a wellshuffled standard 52card deck....
 1.59: A parking lot contains 100 cars, k of which happen to be lemons. We...
 1.60: A wellshuffled 52card deck is dealt to 4 players. Find the probab...
 1.61: Hypergeometric probabilities. An urn contains n balls, out of which...
 1.62: Correcting the number of permutations for indistinguishable objects...
Solutions for Chapter 1: Sample Space and Probability
Full solutions for Introduction to Probability,  2nd Edition
ISBN: 9781886529236
Solutions for Chapter 1: Sample Space and Probability
Get Full SolutionsSince 62 problems in chapter 1: Sample Space and Probability have been answered, more than 7362 students have viewed full stepbystep solutions from this chapter. Chapter 1: Sample Space and Probability includes 62 full stepbystep solutions. Introduction to Probability, was written by and is associated to the ISBN: 9781886529236. This textbook survival guide was created for the textbook: Introduction to Probability,, edition: 2. This expansive textbook survival guide covers the following chapters and their solutions.

2 k p  factorial experiment
A fractional factorial experiment with k factors tested in a 2 ? p fraction with all factors tested at only two levels (settings) each

Bayes’ theorem
An equation for a conditional probability such as PA B (  ) in terms of the reverse conditional probability PB A (  ).

Central tendency
The tendency of data to cluster around some value. Central tendency is usually expressed by a measure of location such as the mean, median, or mode.

Coeficient of determination
See R 2 .

Comparative experiment
An experiment in which the treatments (experimental conditions) that are to be studied are included in the experiment. The data from the experiment are used to evaluate the treatments.

Conditional probability density function
The probability density function of the conditional probability distribution of a continuous random variable.

Continuity correction.
A correction factor used to improve the approximation to binomial probabilities from a normal distribution.

Control chart
A graphical display used to monitor a process. It usually consists of a horizontal center line corresponding to the incontrol value of the parameter that is being monitored and lower and upper control limits. The control limits are determined by statistical criteria and are not arbitrary, nor are they related to speciication limits. If sample points fall within the control limits, the process is said to be incontrol, or free from assignable causes. Points beyond the control limits indicate an outofcontrol process; that is, assignable causes are likely present. This signals the need to ind and remove the assignable causes.

Convolution
A method to derive the probability density function of the sum of two independent random variables from an integral (or sum) of probability density (or mass) functions.

Correction factor
A term used for the quantity ( / )( ) 1 1 2 n xi i n ? = that is subtracted from xi i n 2 ? =1 to give the corrected sum of squares deined as (/ ) ( ) 1 1 2 n xx i x i n ? = i ? . The correction factor can also be written as nx 2 .

Correlation matrix
A square matrix that contains the correlations among a set of random variables, say, XX X 1 2 k , ,…, . The main diagonal elements of the matrix are unity and the offdiagonal elements rij are the correlations between Xi and Xj .

Cumulative distribution function
For a random variable X, the function of X deined as PX x ( ) ? that is used to specify the probability distribution.

Density function
Another name for a probability density function

Expected value
The expected value of a random variable X is its longterm average or mean value. In the continuous case, the expected value of X is E X xf x dx ( ) = ?? ( ) ? ? where f ( ) x is the density function of the random variable X.

Exponential random variable
A series of tests in which changes are made to the system under study

Factorial experiment
A type of experimental design in which every level of one factor is tested in combination with every level of another factor. In general, in a factorial experiment, all possible combinations of factor levels are tested.

Firstorder model
A model that contains only irstorder terms. For example, the irstorder response surface model in two variables is y xx = + ?? ? ? 0 11 2 2 + + . A irstorder model is also called a main effects model

Fisher’s least signiicant difference (LSD) method
A series of pairwise hypothesis tests of treatment means in an experiment to determine which means differ.

Fixed factor (or fixed effect).
In analysis of variance, a factor or effect is considered ixed if all the levels of interest for that factor are included in the experiment. Conclusions are then valid about this set of levels only, although when the factor is quantitative, it is customary to it a model to the data for interpolating between these levels.

Harmonic mean
The harmonic mean of a set of data values is the reciprocal of the arithmetic mean of the reciprocals of the data values; that is, h n x i n i = ? ? ? ? ? = ? ? 1 1 1 1 g .