Binomial probability distribution: Binomial probability distribution Presented by: Hania Shah Nadia haroot Uzaima Sadia haroot shazia
What is Probability? : What is Probability? A measurement of uncertainty is called probability. Example: Suppose the morning news indicates that there is a good chance for rain today, how does one decide whether or not to carry an umbrella? The decision will depend, in part, on what chance or probability is attached to the event “that it will rain today”.
Definitions: Definitions Experiment: Any well defined process from which observations (data) can be obtained The sample space: A simple event is an outcome resulting from a single trial of an experiment. Or A set of all possible outcomes of a random experiment is called a sample space. Notation: N n Example: Sample space of a coin tossing three times S= { (HHH), (HHT), (HTH), (THH), (HTT), (THT), (TTH), (TTT)} Here N = 2 And n = 3 So 2 3 = 8
PowerPoint Presentation: Random Variable: A random variable is defined by a function whose value is a real number determined by each outcome in the sample space. A variable (typically represented by x ) that has a single numerical value, determined by chance, for each outcome of a procedure Denoted by capital alphabets i.e.; X ,Y, Z.
PowerPoint Presentation: Binomial Experiment: A binomial experiment (also known as a Bernoulli trial) is a statistical experiment that has the following properties: The experiment consists of n repeated trials. Each trial can result in just two possible outcomes. We call one of these outcomes a success and the other, a failure. The probability of success, denoted by p , is the same on every trial. The trials are independent; that is, the outcome on one trial does not affect the outcome on other trials. The number of successes X in n trials of a binomial experiment is called a binomial random variable. The probability distribution of the random variable X is called a binomial distribution, and is given by the formula: f(x)=P(X=x)= n Cr.p x . q n -x
Parameters of binomial distribution: Parameters of binomial distribution There are three parameters of binomial. n P Range of parameters: n= 0,1,2,3…… p = 0 to 1
Example : Example Consider the following statistical experiment. You flip a coin 2 times and count the number of times the coin lands on heads. This is a binomial experiment because: The experiment consists of repeated trials. We flip a coin 2 times. Each trial can result in just two possible outcomes - heads or tails. The probability of success is constant - 0.5 on every trial. The trials are independent; that is, getting heads on one trial does not affect whether we get heads on other trials.
The binomial distribution: The binomial distribution In probability theory and statistics, the binomial distribution is the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments each of which yields success with probability p, such a experiment is also called a Bernoulli experiment or Bernoulli trial. Bernoulli trial: When n=1 the binomial distribution is the Bernoulli distribution.
PowerPoint Presentation: Binomial Probability The binomial probability refers to the probability that a binomial experiment results in exactly x successes. For example, in the above table, we see that the binomial probability of getting exactly one head in two coin flips is 0.50. Given x , n , and P , we can compute the binomial probability based on the following formula: Binomial Formula. Suppose a binomial experiment consists of n trials and results in x successes. If the probability of success on an individual trial is P , then the binomial probability is: b( x ; n, P ) = n C x * P x * (1 - P) n – x
Combination by factorial: Combination by factorial for k = 0, 1, 2, ..., n , where
PowerPoint Presentation: Notation: The following notation is helpful, when we talk about binomial probability. x : The number of successes that result from the binomial experiment. n : The number of trials in the binomial experiment. p : The probability of success on an individual trial. q : The probability of failure on an individual trial. (This is equal to 1 - p ) b( x ; n, p ): Binomial probability - the probability that an n -trial binomial experiment results in exactly x successes, when the probability of success on an individual trial is P .
PowerPoint Presentation: Example 1 Suppose a die is tossed 5 times. What is the probability of getting exactly 2 fours? Solution: This is a binomial experiment in which the number of trials is equal to 5, the number of successes is equal to 2, and the probability of success on a single trial is 1/6 or about 0.167. Therefore, the binomial probability is: b(2; 5, 0.167) = 5 C 2 * (0.167) 2 * (0.833) 3 b(2; 5, 0.167) = 0.161
The Binomial Probability Distribution: The Binomial Probability Distribution An experiment has a binomial probability distribution if three conditions are satisfied. a. There are a fixed number of trials. The number of trials is denoted by n . b. The trials are independent. c. The only outcomes of this experiment can be classified as "succeed" or "fail" (equivalently "yes" or "no"). Furthermore, the probability of success is fixed. The probability of success is denoted by p .
PowerPoint Presentation: Cumulative Binomial Probability A cumulative binomial probability refers to the probability that the binomial random variable falls within a specified range (e.g., is greater than or equal to a stated lower limit and less than or equal to a stated upper limit). For example the cumulative binomial probability of obtaining 45 or fewer heads in 100 tosses of a coin. b(x < 45; 100, 0.5) = b(x = 0; 100, 0.5) + b(x = 1; 100, 0.5) + ... + b(x = 44; 100, 0.5) + b(x = 45; 100, 0.5)
PowerPoint Presentation: Example 1 Image source A die is tossed 3 times. What is the probability of (a) No fives turning up? (b) 1 five? (c) 3 fives? Solution: This is a binomial distribution because there are only 2 possible outcomes (we get a 5 or we don't).
PowerPoint Presentation: Now, n =3 for each part. Let X = number of fives appearing. (a) Here, x = 0. P ( X =0) = Cnxpxqn − x = C 30(16)0(56)3 =0.5787 (b) Here, x = 1. P ( X =1) = Cnxpxqn − x = C 31(16)1(56)2 =0.34722 (c) Here, x = 3. P ( X =3)= Cnxpxqn − x = C 33(16)3(56)0 =4.6296×10−3
PowerPoint Presentation: Example 2 Hospital records show that of patients suffering from a certain disease, 75% die of it. What is the probability that of 6 randomly selected patients, 4 will This is a binomial distribution because there are only 2 outcomes (the patient dies, or does not). recover? Sol: Let X = number who recover
PowerPoint Presentation: Here, n =6 and x =4. Let p =0.25 (success, that is, they live), q =0.75 (failure, i.e. they die). The probability that 4 will recover: P ( X ) = Cnx pxqn − x = C 64(0.25)4(0.75)2 =15×2.1973×10−3 =0.0329595
PowerPoint Presentation: A manufacturer of metal pistons finds that on the average, 12% of his pistons are rejected because they are either oversize or undersize. What is the probability that a batch of 10 pistons will contain no more than 2 rejects? (b) at least 2 rejects? Solution: (In this case, "success" means rejection!) Here, n =10, p =0.12, q =0.88. (a) No rejects. That is, when x =0: P ( X ) = Cnxpxqn − x = C 100(0.12)0(0.88)10 =0.2785
PowerPoint Presentation: One reject. That is, when x =1 P ( X ) = C 101(0.12)1(0.88)9 =0.37977 Two rejects. That is, when x =2: P ( X ) = C 102(0.12)2(0.88)8 =0.23304 So the probability of getting no more than 2 rejects is: Probability = P ( X ≤2) =0.2785+0.37977+0.23304 =0.89131
PowerPoint Presentation: (b) We could work out all the cases for X =2,3,4,...,10, but it is much easier to proceed as follows: Probablity of at least 2 rejects =1− P ( X ≤1) =1−( P ( x 0)+ P ( x 1)) =1−(0.2785+0.37977) =0.34173