Elementary Probability PDF

University of Algiers 1 Faculty of pharmacy Department of industrial pharmacy Biostatistics Elementary Probability Dr.KADRI [email protected] Contents I-Introduction II-Basic definitions III- Probability IV- Laws of Probability V- Parameters of a distributi...

University of Algiers 1 Faculty of pharmacy Department of industrial pharmacy Biostatistics Elementary Probability Dr.KADRI [email protected] Contents I-Introduction II-Basic definitions III- Probability IV- Laws of Probability V- Parameters of a distribution VI- Type of probability distribution I-Introduction  In general, probability is the measure of how likely an outcome is to occur , we cannot predict with certainty how the experiment will turn out, rather we can only list the collection of possible outcomes. II- BASIC DEFINITIONS  Random Experiment ꞉ is an experiment whose outcome cannot be predicted with certainty, before the experiment is run.  We usually assume that the experiment can be repeated infinitely under essentially the same conditions Example  Throwing a dice  Throwing a coin  II- BASIC DEFINITIONS  Random Variable꞉ the variables whose exact value can not be predicted  Two types of random variables  Discrete random variables (countable set of possible outcomes) Exp :The number of children in a family  Continuous random variable (unbroken chain of possible outcomes) Exp : Leaf areas of the 4 th leaf of a given plant II- BASIC DEFINITIONS  Event : an outcome or set of outcomes Let A be an event that two or more heads appear consecutively from an experiment of throwing a coin three times Then A={ HHH, HHT, THH}.  II- BASIC DEFINITIONS  Sample space : the set of all possible outcomes of a chance experiment it is denoted by Ω. Exp: Tossing a coin once, Ω = {H, T}. Rolling a dice once, Ω = {1, 2, 3, 4, 5, 6}.  III- probability  Probability is the likelihood or chance that a particular event will occur. Exp : Chance of picking a black card from a deck of cards P = number of possible points of the event/ number of all possible points  III- probability  Example 1 : A spinner has 4 equal sectors colored yellow, blue, green and red. After spinning the spinner, what is the probability of landing on each color? The possible outcomes of this experiment are yellow, blue, green, and red.  III- probability  Example 2 : What is the probability of rolling a 6 with a well-balanced dice?  The probability is 1/6 = 0.167  Example 3 : suppose that out of N =100;000 persons , a total of 5500 are positive reactors to a certain screening test , what is the probability of being positive? III- probability Axioms of Probability III- probability  The probability is always between 0 and 1  Impossible Event : an event that has no chance of occurring (probability = 0)  Certain Event : an event that is sure to occur (probability = 1)  Equally likely out comes: If each out come in an experiment has the same chance to occur III- probability  Mutually exclusive events : Events that cannot occur simultaneously P(A∪B) = P(A) + P(B) If we toss a coin, either heads or tails might turn up, but not heads and tails at the same time. III- probability Non mutually exclusive events : Events that can occur simultaneously P(A or B ) = P(A ) + P(B) - P(A ∩ B ) III- probability  Complement of an event A: All events that are not part of event A P(X) = 1 − P(A) Example : All days from 2015 that are not in January  Independent events: Two or more events are independent if the occurrence or non-occurrence of either of one does not affect the probability of the occurrence of the other. P(A ∩ B) = P(A) × P(B) Choosing a marble from a jar AND and landing on heads after tossing a coin III- probability Example : P(C) = 0.5, P (D) = 0.3, P(C∩D) = 0.10 Are C and D independent? Solution: P(C ∩ D) = P(A) × P(D) = 0.5 × 0.3 = 0.15 P(C ∩ D) 6= 0.10 Hence, C and D are dependent  IV-Laws of Probability  Probability Distribution of a discrete random variable :The probability distribution of a discrete random variable is a list of probabilities associated with each of its possible values. It is also sometimes called the probability function or the probability mass function. f (x) given by f (x) = P(Ax ), This is also denoted by P(X = x).  The distribution f satisfies the conditions  IV-Laws of Probability  The table given below represent the probability distribution of X  IV-Laws of Probability Figure 2. Probability distribution of Y  IV-Laws of Probability  Cumulative Probability: It is defined as the probability of observing less than or equal a given number of success.  Probability distribution and cumulative probability distribution IV-Laws of Probability Figure 3. Cumulative distribution of Y IV-Laws of Probability  Probability Distribution of continuous random variables : Random variable X is continuous if there is an integrable function  The function fX is called the probability density function (pdf)  Continuous random variable is not defined at specific values. Instead, it is defined over an interval of values, and is represented by the area under a curve. IV-Laws of Probability Areas Under the Curve AREAS = probabilities Top figure: histogram, ages ≤ 9 shaded Bottom figure: pdf, ages ≤ 9 shaded Both represent proportion of population ≤ 9 V- Parameters of a distribution Expected value (mean) - µ : the sum of the probability of each possible outcome of the experiment multiplied by the outcome value E(X)= µ = ∑ pi xi V- Parameters of a distribution Example Consider the distribution of X bellow , calculate the mean E(X) = µ = 1(1/36) + 2(3/36) + 3(5/36) + 4(7/36) + 5(9/36) + 6(11/36) = 4.47 V- Parameters of a distribution Variance of a distribution ( ) – Var(X) var(a) = 0 var(ax) = a² var(X) Var(X+Y) = var(X) + var(Y)  V- Parameters of a distribution Example  V- Parameters of a distribution  Standard deviation : is defined as the square root of variance. This indicator has more benefits than the variance in interpreting results. VI- Type of probability distribution Discrete probability distribution Bernoulli distribution : describes a simple random experiment with two possible outcomes: success (1) and failure (0) P{X =1} = p P{X = 0} = q = 1− p Mean value: E[X] = q⋅0 + p⋅1 = p Variance: Var(X)=pq= VI- Type of probability distribution Discrete probability distribution  Binomial distribution : The experiment of Bernoulli carried out n times. n! = 1*2*3*…*n and 0! =1 VI- Type of probability distribution Binomial distribution Where, p(x=k) = Probability of k successes in n trials p = Probability of success q = Probability of failure= 1-p. k = Number of successes desired. n = Number of trials undertaken VI- Type of probability distribution  Binomial distribution For proportion (% ) Mean = μ = (np)/n= p Var(% ) = p(1-p)/n  VI- Type of probability distribution  Binomial distribution  Using Binomial Tables  Often a number of binomial probabilities need to be evaluated for the same n and p, which would be tedious if each probability had to be calculated  VI- Type of probability distribution  Binomial distribution Example: A fair coin is tossed 8 times. What is the probability of obtaining 6 or more heads? Solution: When a fair coin is tossed, the probabilities of head and tail in case of an unbiased coin are equal, p = q = ½ or 0.5. According to formula The probabilities of obtaining 6 heads is :  VI- Type of probability distribution  Binomial distribution Probability of obtaining 7 head is 0.031 Probability of obtaining 8 head is 0.0039 Therefore the probability of obtaining 6 heads or more heads is= 0.105+0.031+0.0039= 0.1399  VI- Type of probability distribution  Binomial distribution Example : Evaluate the probability of 2 lymphocytes out of 10 white blood cells if the probability that any one cell is a lymphocyte is 0.2. Solution: Refer to Table 1 with n = 10, k = 2, p =.20. The appropriate probability, given in the k = 2 row and p = 0.20 column under n = 10, is 0.3020  VI- Type of probability distribution  Binomial distribution Example : An investigator notices that the number of children who develop a chronic bronchitis in the first year of life is 3 of 20 households , as compared with the national incidence of chronic bronchitis, which is 5% in the first year of life , how likely are infants at least 3 of 20 households to develop chronic bronchitis?  VI- Type of probability distribution  Binomial distribution These three probabilities in the sum can be evaluated using the binomial table. Refer to n = 20, p = 0.05, P(X = 0) =0.3585 P(X = 1) = 0.3774 P(X = 2) =0.1887. Thus P (X ≥ 3) = 1 − (0.3585 + 0.3774 + 0.1887) = 0.0754 VI- Type of probability distribution  Binomial distribution When the number of trials n is from moderate to large (n > 30) we approximate the binomial distribution by a normal distribution VI- Type of probability distribution Poisson Distribution : The binomial distribution with n very large and at the same time p very small. : The mean