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Math 211 Introduction to Statistics Chapter 5 Probability and Some Probability Distributions Key words: Sample space, sample point, tree diagram, events, complement, union and intersection of an event, mutually exclusive events; Counting techniques: multiplication rule, permutation, combination, probability of an event; Additive rules; Conditional probability, independent events, multiplicative rules, Bayes’ rule. SAMPLE SPACE Sample Space. The set of all possible outcomes of a statistical experiment is called a Sample space. It is represented by the symbol S. Sample Point. Each outcome in a sample space is called an element or a sample point of the sample space. Example 1. (a) Consider the experiment of tossing a coin. The sample space S of possible outcomes may be written as S= H , T . (b) Consider the experiment of flipping a die. Then the elements of the sample space S is listed as S= 1, 2,3, 4,5,6 . (c) Now consider the experiment of tossing a die and then a coin once. The resultant sample space can be obtained using TREE DIAGRAM. That is, Therefore the sample sample S is S= 1H ,1T , 2H , 2T ,3H ,3T , 4H , 4T ,5H ,5T ,6 H ,6T . Sonuc Zorlu 1 Lecture Notes Math 211 Introduction to Statistics Event. An event is a subset of a sample space. For example A 4H ,6T is an event defined on S. One can define 212 events on S. Empty set , is an impossible event and S is a sure event. Any subset of S is represented by capital letters such as A, B, C… The Complement of an Event. The complement of an event A with respect to S is the subset of all elements of S that are not in A. The complement of A is denoted by the symbol A ' or Ac . The Intersection of Events. The intersection of two events A and B, denoted by the symbol A B is the event containing all elements that are common to A and B. Mutually Exclusive Events. Two events A and B are mutually exclusive or disjoint if A B , that is if A and B have no common elements in common. The Union of Events. The union of two events A and B, denoted by the symbol A B is the event containing all elements that belong to A or B or both. Important Notes. The following results may easily be verified by means of Venn diagrams. (1) A (2) A A (3) A A ' (4) A A ' S (5) S ' (6) ' S (7) A ' ' A (8) (9) A B ' A ' B ' 1st De Morgan Rule A B ' A ' B ' 2nd De Morgan Rule Example 2. If S 0,1, 2,3, 4,5,6,7,8,9 and A 0, 2, 4,6,8 , B 1,3,5,7,9 , C 2,3, 4,5 and D 1,6,7 , list the elements of the sets corresponding to the following events: (a) A C S (b) A B ( A and B are mutually exclusive events) (c) C ' 0,1,6,7,8,9 (d) C ' D B 0,1, 6, 7,8,9 1, 6, 7 1,3,5, 7,9 1, 6, 7 1,3,5, 7,9 1, 7 (e) S C ' C ' 0,1,6,7,8,9 A C D ' 2, 4 0, 2,3, 4,5,8,9 (f) 2, 4 AC Sonuc Zorlu 2 Lecture Notes Math 211 Introduction to Statistics COUNTING SAMPLE POINTS Multiplication Rule. If an operation can be performed in n1 ways, and if for each of these a second operation can be performed in n2 ways, then the two operations can be performed together in n1n2 ways. Example 1. How many breakfasts consisting of a drink and a sandwich are possible if we can select from 3 drinks and 4 kinds of sandwiches? Solution : n1 3 and n2 4 , therefore there are n1xn2 3x4 12 different ways to choose a breakfast. Example 2. A certain shoe comes in 5 different styles with each style available in 4 distinct colors. If the store wishes to display pairs of these shoes showing all of its various styles and colors, how many different pairs would the store have on display? Solution : 5 4 20 different pairs are available. Example 3. In how many ways can a true-false test consisting of 9 questions be answered? Solution : There are 2 2 2 2 2 2 2 2 2 29 different answers for this test. Example 4. How many even three-digit numbers can be formed fro the digits 1,2,5,6, and 9 if (a) each digit can be used only once? (b) each digit can be repeated? Solution : (a) Since the number must be even we have only 2 choices for the units position. For each of these there are 4 choices for hundreds position and 3 choices for tens positions. Therefore we can form a total of 2 4 3 24 even three-digit numbers. (b) Similarly for the units position there are 2 choices, since the numbers can be repeated we have 4 choices for the hundreds and tens positions. Hence, totally there are 2 4 4 32 even three-digit numbers. Permutation . A permutation is an arrangement of all or part of a set of objects. The number of permutations of n distinct objects is n! . Theorem . The number of permutations of n distinct objects taken r at a time is n! n Pr n r ! Sonuc Zorlu 3 Lecture Notes Math 211 Introduction to Statistics Example 5 . In how many ways can a Society schedule 3 speakers for 3 different meetings if they are available on any of 5 possible dates? Solution : The total number of possible schedules is 5! 5.4.3 60 . 5 P3 5 3 ! Theorem . The number of permutations of n distinct objects arranged in a circle is n 1! . Theorem . The number of distinct permutations of n things of which n1 are of one kind, n2 of a second kind,…, nk of a k th kind is n! n1 !n2 !...nk ! Example 6. How many different ways can 3 red, 4 yellow, and 2 blue bulbs be arranged in a string of Christmas tree lights with 9 sockets? Solution : The total number of distinct arrangements is 9! 1260 . 3!4!2! Example 7. Suppose that 10 employees are to be divided among three jobs with 3 employees going to job I, 4 to job II, and 3 to job III. In how many ways can the job assignment be made? Solution. 10! =2200 3!4!3! Theorem. The number of combinations of n distinct objects taken r at a time is n n! n Cr r (n r )!r ! Example 8. From 4 mathematicians and 6 computer scientists, find the number of committees that can be formed consisting of 2 mathematicians and 4 computer scientists. 4 6 4! 6! 4.3.2!6.5.4! Solution. . 3.6.5 90 2!2.2.4! 2 4 2!2! 2!4! Sonuc Zorlu 4 Lecture Notes Math 211 Introduction to Statistics Probability of an Event Definition. Suppose that an experiment has associated with it a sample space S. A probability is a numerically valued function that assigns a number P A to every event A so that the following axioms hold: (1) P( A) 0 (2) P( S ) 1 (3) If A1 , A2 ,..., is a sequence of mutually exclusive events (i.e. Ai Aj for any i j ), then P Ai P( Ai ) i 1 i 1 Example 1. A coin is tossed twice. What is the probability that at least one head occurs? The sample space for this experiment is S HH , HT , TH , TT . If the coin is balanced, each of these outcomes would be equally likely to occur. Therefore we assign a probability w to each sample point. Then 4w=1 or w=1/4.If A represents the event of at least one head occurring, then 1 1 1 3 A = {HH , HT , TH } and P( A)= + + = . 4 4 4 4 Theorem . If an experiment can result in any one of N different equally likely outcomes, and if exactly n of these outcomes correspond to event A , then the probability of event A is n P( A) = N Additive Rules Theorem. If A and B are two events, then P (A È B) = P( A) + P( B) - P (A Ç B) Corollary. If A and B are mutually exclusive events, then P (A È B) = P( A) + P( B) Corollary. If A1 , A2 , A3 ,..., An are mutually exclusive events, then P (A1 È A2 È ... È An ) = P( A1 ) + P( A2 ) + ... + P( An ) Theorem. For three events A , B and C , P (A È B È C ) = P( A) + P( B) + P (C )- P (A Ç B)- P (A Ç C )- P (B Ç C )+ P (A Ç B Ç C ). Sonuc Zorlu 5 Lecture Notes Math 211 Introduction to Statistics Example 1. What is the probability of getting a total 7 or 11 when a pair of dice are tossed? Solution. Let A : event that 7 occurs ; B : event that 11 occurs , therefore 1 1 2 P (A È B ) = P( A) + P( B) = + = 6 18 9 Theorem. If A and A¢ are complementary events, then P( A) + P( A¢) = 1 Example 2. In a random experiment it is known that P( A) 0.35, P( A B) 0.19 , and P( A B ) 0.15. Calculate P( B) . Exercises Exercise 1. A pair of dice is tossed. Find the probability of getting (a) a total of 8 (b) at most a total of 5. Exercise 2. Two cards are drawn in succession from a deck without replacement. What is the probability that both cards are greater than 3 and less than 9? Exercise 3. If 3 books are picked at random from a shelf containing 5 novels, 3 books of poems, and a dictionary, what is the probability that (a) the dictionary is selected? (b) 2 novels and 1 book of poems are selected? Exercise 4. If 2 of the 10 employees are female and 8 male, what is the probability that exactly one female gets selected among the three? Exercise 5. A package of 6 light bulbs contain 2 defective bulbs. If 3 bulbs are selected for use, find the probability that none is defective. Exercise 6. How many four-digit serial numbers can be formed if no digit is to be repeated within any one number? Exercise 7. Seven applicants have applied for two jobs. How many ways can the jobs be filled if (a) the first person chosen receives a higher salary than the second? (b) There are no differences between the jobs? Exercise 8. From a group of 4 men and 5 women, how many committees of size 3 are possible (a) with no restrictions? (b) with 1 man and 2 women? (c) with 2 men and 1 woman if a certain man must be on the committee? Exercise 9. In how many ways can the letters of the word ‘KRAKATOA’ be arranged? Sonuc Zorlu 6 Lecture Notes Math 211 Introduction to Statistics Random Variables and Probability Distributions In statistics we deal with random variables- variables whose observed value is determined by chance. Random variables usually fall into one of two categories: discrete or continuous Random Variable. A random variable (r.v.) is a function that associates a real number with each element in the sample space. Random variables will be denoted by uppercase letters and their observed numerical values by lowercase letters. Discrete Random Variable. A random variable is discrete if it can assume at most a finite or a countably infinite number of possible values. Example 1. Two balls are drawn in succession without replacement from an urn containing 4 red and 3 black balls. The possible outcomes and values y of the random variable Y where Y is the number of red balls are: Sample Space y RR 2 RB 1 BR 1 BB 0 Continuous Random Variable. A random variable is continuous if it can assume any value in some interval or intervals of real numbers and the probability that it assumes any specific value is 0. Discrete Probability Distributions Definition. The set of ordered pairs x, f x is a probability function, probability mass function or probability distribution of the discrete random variable X if, (1) f x 0 (2) f x 1 x (3) P X x f x . Example 1. A committee of size 5 is to be selected at random from 3 chemists and 5 mathematicians. Find the probability distribution (p.d.) for the number of chemists on the committee. Let X be the number of chemists on the committee. Then x : 0,1, 2,3 . Sonuc Zorlu 7 Lecture Notes Math 211 Introduction to Statistics 3 5 0 5 1 ; P X 0 f 0 56 8 5 3 5 1 4 15 P X 1 f 1 56 8 5 3 5 3 5 2 3 30 3 2 10 ; P X 2 f 2 P X 3 f 3 56 56 8 8 5 5 Therefore the probability distribution of X is x f x 0 1 56 1 15 56 2 30 56 3 10 56 Exercise 1. Among 10 applicants for an open position 6 are female and are males. Suppose 3 applicants are randomly selected from the applicant pool for final interviews. Find the probability distribution for X , which is the number of female applicants among the final three. Exercise 2. Let w be a random variable giving the number of heads minus the number of tails in three tosses of a coin. (a) List the elements of the sample space (b) Assign a value w of W to each sample points. (c) Find the probability distribution of the random variable W assuming that the coin is biased so that a head is twice as likely to occur as a tail. Cumulative Distribution. The cumulative distribution F x of a discrete random variable X with probability distribution f x is F x P X x f t , for x . tx Example 2. Find the cumulative distribution of the number of red balls in example 1. Using 10 F x , show that f 3 . 56 F 0 f 0 Sonuc Zorlu 1 ; 56 8 Lecture Notes Math 211 Introduction to Statistics F 1 f 0 f 1 16 56 46 ; 56 F 3 f 0 f 1 f 2 f 3 1 . F 2 f 0 f 1 f 2 Hence, 0, 1 56 16 F x 56 46 56 1 46 10 . Now, f 3 F 3 F 2 1 56 56 if x 0 if 0 x 1 if 1 x 2 if 2 x 3 if x 3 Continuous Probability Distributions Definition. (Probability Density Function) The function f x is a probability density function for the continuous random variable X , defined over the set of real numbers , if (1) f x 0, for all x (2) f x dx 1 b (3) P a X b f x dx . a Note that for a continuous random variable X , a P X a f x dx 0 . a Cumulative Distribution. The cumulative distribution F x of a continuous random variable X with density function f x is F x P X x x f t dt for x . Sonuc Zorlu 9 Lecture Notes Math 211 Introduction to Statistics Example 1. Suppose that a random variable X has a probability density function given by kx 1 x 0 x 1 f x elsewhere 0 (a) Find the value of k that makes this a probability function. 1 kx 1 x dx 1 k 6 0 (b) Find P 0.4 X 1 0.4 2 0.4 3 6 x 1 x dx 1 6 2 3 0.332 0.4 (c) Find F x P X x and sketch the graph of this function. 1 if x 0 0, 2 F ( x) 3 x 2 x 3 , if 0 x 1 1, if x 1 Some Useful Probability Distributions The observations generated by different statistical experiments have the same general type of behavior. Random variables associated with these experiments can be described by essentially the same probability distribution and therefore can be represented by a single formula. The followings are the probability distributions that will be covered in this chapter: Binomial Distribution Poisson Distribution and Poisson Process Normal Distributions The Binomial Distribution Perhaps the most commonly used discrete probability distribution is the binomial distribution. An experiment which follows a binomial distribution will satisfy the following requirements (think of repeatedly flipping a coin as you read these): 1. The experiment consists of n identical trials, where n is fixed in advance. 2. Each trial has two possible outcomes, S or F, which we denote ``success'' and ``failure'' and code as 1 and 0, respectively. 3. The trials are independent, so the outcome of one trial has no effect on the outcome of another. 4. The probability of success, p is constant from one trial to another. Sonuc Zorlu 10 Lecture Notes Math 211 Introduction to Statistics The random variable X of a binomial distribution counts the number of successes in n trials. The probability that X is a certain value x is given by the formula n n x P X x b x, n, p p x 1 p x n where 0 p 1 and x 0,1, 2,...., n . Recall that the quantity , ``n choose x,'' above is x We could use the formulas previously given to compute the mean and variance of X. However, for the binomial distribution these will always be equal to E X np and Var X 2 npq NOTE:\ A particularly important example of the use of the binomial distribution is when sampling with replacement (this implies that p is constant). Example 1. Suppose we have 10 balls in a bowl, 3 of the balls are red and 7 of them are blue. Define success S as drawing a red ball. If we sample with replacement, P(S)=0.3 for every trial. Let's say n=20, then X b x, 20,0.3 and we can figure out any probability we want. For example, The mean and variance are Example 2. The probability that a patient recovers from a rare blood disease is 0.4. If 15 people are known to have contracted this disease, what is the probability that (a) at least 10 survive? (b) from 3 to 7 survive? (c) exactly 5 survive? Solution. Probability of success= p 0.4 , and the probability of failure= q 0.6 . n 15 and X : no. of surviving patients (a) P X 10 1 P X 9 1 B 9;15,0.4 1-0.9662=0.0338 (b) P 3 X 7 P 4 X 6 b 4;15,0.4 b 5;15,0.4 b 6;15,0.4 0.509 15 5 155 0.186 (c) P X 5 b 5;15, 0.4 0.4 0.6 5 Sonuc Zorlu 11 Lecture Notes Math 211 Introduction to Statistics Example 3. A traffic control engineer reports that 75% of the vehicles passing through a checkpoint are from within the state. What is the probability that fewer than 2 of the next 9 vehicles are from out of the state? Solution. Probability of success= p 0.25 , and the probability of failure= q 1 0.25 0.75 . n 9 and X : no. of vehicles passing through the checkpoint P X 2 P X 1 b 0;9, 0.25 b 1;9, 0.25 9 9 0 9 1 8 0.25 0.75 0.25 0.75 0.3 0 1 Example 4. Assuming that 6 in 10 automobile accidents are due mainly to speed violation, (a) find the probability that among 8 automobile accidents 6 will be due mainly to a speed violation. (b) Find the mean and variance of the number of automobile accidents for 8 automobile accidents. Solution. Probability of success= p 6 /10 , and the probability of failure= q 1 0.25 0.75 . and X : no. of automobile accidents n 8 6 8 6 8 6 6 (a) P X 6 b 6;8, 6 /10 1 0.2090 6 10 10 (b) The mean of the number of automobile accidents is E X np 8 * The variance of the no. of auto. accidents is 2 Var X npq 8 * 6 4.8 . 10 6 4 * 1.92 . 10 10 The Poisson Distribution The Poisson distribution is most commonly used to model the number of random occurrences of some phenomenon in a specified unit of space or time. For example, The number of phone calls received by a telephone operator in a 10-minute period. The number of flaws in a bolt of fabric. The number of typos per page made by a secretary. For a Poisson random variable, the probability that X is some value x is given by the formula P X x f x; t e t t x! x x 0,1, 2,... where is the average number of occurrences per unit time or region denoted by t . For the Poisson distribution, Sonuc Zorlu 12 Lecture Notes Math 211 Introduction to Statistics E X t and Var X t . Example 1. The number of false fire alarms in a suburb of Houston averages 2.1 per day. Assuming that a Poisson distribution is appropriate, the probability that 4 false alarms will occur on a given day is given by Example 2. During a laboratory experiment the average number of radioactive particles passing through a counter in 1 millisecond is 4. What is the probability that 6 particles enter the counter in a given millisecond? Example 3. A secretary makes 2 errors per page, on average. What is the probability that on the next page he or she will make (a) 4 or more errors? (b) no errors? Example 4. The number of customers arriving per hour at a certain automobile service facility is assumed to follow a Poisson distribution with 7 . (a) Compute the probability that more than 10 customers will arrive in a 3-hour period. (b) What is the mean number of arrivals during a 4-hour period? Example 5. A restaurant chef prepares tossed salad containing, on average, 5 vegetables. Find the probability that the salad contains more than 5 vegetables (a) on a given day (b) on 3 of the next 4 days (c) for the first time in April on April 5. Poisson Approximation to Binomial Distribution Theorem. Let X be a binomial random variable with probability distribution b x; n, p . When n , p 0 and np remains constant, b x; n, p p x; . Example 1. The probability that a person will die from a certain respiratory infection is 0.002. Find the probability that fewer than 5 of the next 2000 so infected will die. X : number of infected people who will die Given n 2000, p 0.002 , np 2000 * 0.002 4 , 4 P X 5 P X 4 b x; 2000, 0.002 n l arg e , p small 4 x 1 Sonuc Zorlu 13 p x; 4 P 4; 4 by table A 2 0.6288. x 1 Lecture Notes Math 211 Introduction to Statistics Example 2. It is known that 5% of the books bound by a certain bindery have defective bindings. Find the probability that 2 of 100 books bound by this bindery will have defective bindings using (a) the formula for the binomial distribution (b) the Poisson approximation to the Binomial distribution. Solution. (a) Given p 0.05, n 100 , and X , number of defective bindings, 100 2 98 P X 2 b 2;100, 0.05 0.05 0.95 0.081 2 e5 52 (b) P X 2 b 2;100, 0.05 p 2;5 0.084 where np 100 * 0.05 5 . 2! The Normal Distribution The most important continuous probability distribution in the entire field of statistics is the normal distribution. Normal distributions are a family of distributions that have the same general shape. They are symmetric with scores more concentrated in the middle than in the tails. Normal distributions are sometimes described as bell shaped which are shown below. Notice that they differ in how spread out they are. The area under each curve is the same. The height of a normal distribution can be specified mathematically in terms of two parameters: the mean (μ) and the standard deviation (σ). Definition. The density function of the normal random variable X , with mean and variance 2 , is 2 1 1/ 2 x / N x; , e , x 2 where 3.14159.... and e 2.71828 . Sonuc Zorlu 14 Lecture Notes Math 211 Introduction to Statistics Standard normal distribution The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. Normal distributions can be transformed to standard normal distributions by the formula: where X is a score from the original normal distribution, μ is the mean of the original normal distribution, and σ is the standard deviation of original normal distribution. The standard normal distribution is sometimes called the z distribution. A z score always reflects the number of standard deviations above or below the mean a particular score is. For instance, if a person scored a 70 on a test with a mean of 50 and a standard deviation of 10, then they scored 2 standard deviations above the mean. Converting the test scores to z scores, an X of 70 would be: So, a z score of 2 means the original score was 2 standard deviations above the mean. Note that the z distribution will only be a normal distribution if the original distribution (X) is normal. Note : The following figures give us the areas to the right of some z value, to the left of some z value and between two z values. Sonuc Zorlu 15 Lecture Notes Math 211 Introduction to Statistics Areas under portions of the standard normal distribution are shown to the right. About .68 (.34 + .34) of the distribution is between -1 and 1 while about .96 of the distribution is between -2 and 2. Example 1. Given a standard normal distribution, find the area under the curve that lies (a) to the right of z 1.84 (b) between z 1.97 and z 0.86 Solution. (a) P Z 1.84 1 P Z 1.84 by table A3 1 0.9671 0.0329 . (b) P 1.97 Z 0.86 P Z 0.86 P Z 1.97 by table A3 0.8051 0.0244 0.7807 Example 2. Given a normal distribution with 50 and 10 , find the probability that X assumes a value between 45 and 62. Solution. 62 50 45 50 P 45 X 62 P Z P 0.5 Z 1.2 table(1.2) table(0.5) 10 10 = 0.8849-0.3088 0.5764 Example 3. Given a standard normal distribution, find the value of k such that (a) P Z k 0.0427 (b) P Z k 0.2946 (c) P 0.93 Z k 0.7235 Solution. (a) P Z k 0.0427 table(k ) 0.0427 k 1.72 (b) P Z k 0.2946 P Z k 1 P Z k 0.2946 P Z k 1 0.2946 0.7054 table(k ) 0.7054 k 0.54 (c) P 0.93 Z k 0.7235 table(k ) table(0.93) 0.7235 table(k ) 0.7235 0.1762 0.8997 k 1.28 Sonuc Zorlu 16 Lecture Notes Math 211 Introduction to Statistics Example 4. Given the normally distributed random variable X with X 18 and X2 3 (a) Compute P X 13.74 (b) Compute x satisfying P x X 18 0.4332 . Applications of the normal Distribution Example 1. A certain machine makes electrical resistors having a mean resistance of 40 ohms and a standard deviation of 2 ohms. Assuming that the resistamce follows a normal distribution and can be measured to any degree of accuracy, what percenatge of resistors wil have a resistance exceeding 43 ohms? Solution. Let X be the normal random variable, given 40ohms, 2ohms , X 43 40 P X 43 P P Z 1.5 1 P Z 1.5 2 1 table(1.5) 1 0.9332 0.0668 6.68% Example 2. The average grade for a exam is 74 and the standard deviation is 7. Assuming that the grades follow a normal distribution, what is the probability that a student will get grade of at least 60? Normal Approximation to Binomial Distribution Theorem. If X is a binomial random variable with mean np and 2 npq , then the limiting form of distribution of X np Z bin as n npq is the standard normal distribution N 0,1 . Note. We use the normal approximation to binomial distribution whenever p is not close to 0 and 1. If both np and nq are greater than or equal to 5, the approximation will be good. Example 1. The probability that a patient recovers from a blood disease is 0.4. If 100 people are known to have contracted this disease what is the probability that less than 30 survive? Solution. Let X be the number of surviving people from blood disease. Given n 100 and p 0.40 , np 100 * 0.40 40 , 100 * 0.40 * 0.60 0.24 , Sonuc Zorlu 17 Lecture Notes Math 211 Introduction to Statistics X np 29.5 40 P X bin 30 P X nor 29.5 P P Z 2.14 table(2.14) 0.0162 npq 24 Example 2. A coin is tossed 400 times, use the normal approximation to binomial to find the probability of obtaining (a) between 185 and 210 heads inclusive (b) exactly 205 heads (c) less than 176 or more than 227 heads Example 3. A ablanced die is rolled 180 times. Let be the number of cases when die shows the number 4 on its top face. (a) Find X (b) Find X2 (c) Use normal approximation to binomial to approximate P 35 X 40 . Exercises Exercise1. If scores are normally distributed with a mean of 30 and a standard deviation of 5, what percent of the scores is: (a) greater than 30? (b) greater than 37? (c) between 28 and 34? Exercise 2. Assume a normal distribution with a mean of 90 and a standard deviation of 7. What limits would include the middle 65% of the cases? Exercise 3. If is the standard normal random variable, (a) Calculate P 1.3 Z 1.37 (b) If P a Z 1.12 0.6845 , find the value of a . Exercise 4. A research scientists reports that mice will live an average of 40 months when their diets are sharply restricted and enriched with vitamins and proteins. Assuming that the lifetimes of such mice are normally distributed with a standard deviation of 6.3 months, find the probability that a given mouse will live (a) more than 32 months (b) less than 28 months (c) between 37 and 49 months Sonuc Zorlu 18 Lecture Notes