PROBABILITY AND STATISTICS FOR ENGINEERING Hossein Sameti Department of Computer Engineering Sharif University of Technology Independence and Bernoulli Trials

Sharif University of Technology 2 Independence  A, B independent implies: are also independent. Proof for independence of : Events A and B are independent if

Sharif University of Technology 3 Application  let :  Then  Also  Hence it follows that Ap and Aq are independent events! Example

Sharif University of Technology 4 Some Properties of Independence  Let P(A)=0,  Then:  Thus: P(AB) = P(A) P(B) = 0 The event of zero probability is independent of every other event Other independent events cannot be ME  Since:

Sharif University of Technology 5  If and A i s are independent,  Application in solving number theory problems. Application  A family of events { A i } are said to be independent, if for every finite sub collection Remark Application

Sharif University of Technology 6 Example Each switch remains closed with probability p. (a) Find the probability of receiving an input signal at the output. Three independent parallel switches  Let A i = “Switch S i is closed” and R = “input signal is received at the output”. Then Solution

Sharif University of Technology 7 Example – continued  Since any event and its complement form a trivial partition,  Thus: Another Solution Note :The events A 1, A 2, A 3 do not form a partition, since they are not ME. Moreover,

Sharif University of Technology 8 Example – continued  From Bayes’ theorem  Also, because of the symmetry : Solution (b) Find the probability that switch S1 is open given that an input signal is received at the output.

Sharif University of Technology 9 Repeated Trials  A joint performance of the two experiments with probability models (  1, F 1, P 1 ) and (  2, F 2, P 2 ).  Two models’ elementary events:  1,  2  How to define the combined trio ( , F, P)?   =  1   2  Every elementary event  in  is of the form  = ( ,  ).  Events: Any subset A  B of  such that A  F 1 and B  F 2  F : all such subsets A  B together with their unions and complements.

Sharif University of Technology 10 Repeated Trials  In this model, The events A   2 and  1  B are such that  Since the events A   2 and  1  B are independent for any A  F 1 and B  F 2.  So, for all A  F 1 and B  F 2  Now, we’re done with definition of the combined model.

Sharif University of Technology 11 Generalization  Experiments with  let  Elementary events  Events in are of the form and their unions and intersections.  If s are independent, and is the probability of the event in then

Sharif University of Technology 12 Example  A has probability p of occurring in a single trial.  Find the probability that A occurs exactly k times, k  n in n trials.  The probability model for a single trial : ( , F, P)  Outcome of n experiments is : where and  A occurs at trial # i, if  Suppose A occurs exactly k times in . Solution

Sharif University of Technology 13 Example - continued  If belong to A,  Ignoring the order, this can happen in N disjoint equiprobable ways, so:  Where

Sharif University of Technology 14 Bernoulli Trials  Independent repeated experiments  Outcome is either a “success” or a “failure”  The probability of k successes in n trials is given by the above formula, where p represents the probability of “success” in any one trial.

Sharif University of Technology 15 Example  Toss a coin n times. Obtain the probability of getting k heads in n trials.  “head” is “success” (A) and let  Use the mentioned formula.  In rolling a fair dice for eight times, find the probability that either 3 or 4 shows up five times ?  Thus Example

Sharif University of Technology 16 Bernoulli Trials  Consider a Bernoulli trial with success A, where  Let:  As are mutually exclusive,  so,

Sharif University of Technology 17 Bernoulli Trials  For a given n and p what is the most likely value of k ?  Thus if or  Thus as a function of k increases until  The is the most likely number of successes in n trials. Fig. 2.2

Sharif University of Technology 18 Effects Of p On Binomial Distribution

Sharif University of Technology 19 Effects Of p On Binomial Distribution

Sharif University of Technology 20 Effects Of n On Binomial Distribution

Sharif University of Technology 21 Example  In a Bernoulli experiment with n trials, find the probability that the number of occurrences of A is between k 1 and k 2. Solution

Sharif University of Technology 22 Example  Suppose 5,000 components are ordered. The probability that a part is defective equals 0.1. What is the probability that the total number of defective parts does not exceed 400 ? Solution

Sharif University of Technology 23  If k max is the most likely number of successes in n trials,  or  So,  This connects the results of an actual experiment to the axiomatic definition of p.

Sharif University of Technology 24 Bernoulli’s Theorem  A : an event whose probability of occurrence in a single trial is p.  k : the number of occurrences of A in n independent trials  Then,  i.e. the frequency definition of probability of an event and its axiomatic definition ( p) can be made compatible to any degree of accuracy.

Sharif University of Technology 25 Proof of Bernoulli’s Theorem  Direct computation gives:  Similarly,

Sharif University of Technology 26 Proof of Bernoulli’s Theorem - continued   Equivalently,  Using equations derived in the last slide,

Sharif University of Technology 27 Proof of Bernoulli’s Theorem - continued  Using last two equations,  For a given can be made arbitrarily small by letting n become large.  Relative frequency can be made arbitrarily close to the actual probability of the event A in a single trial by making the number of experiments large enough.  As the plots of tends to concentrate more and more around Conclusion

Sharif University of Technology 28 Example: Day-trading strategy  A box contains n randomly numbered balls (not necessarily 1 through n)  m = np (p<1) of them are drawn one by one by replacement  The drawing continues until a ball is drawn with a number larger than the first m numbers.  Determine the fraction p to be initially drawn, so as to maximize the probability of drawing the largest among the n numbers using this strategy.

Sharif University of Technology 29 Day-trading strategy: Solution  Let be the event of: drawn ball has the largest number among all n balls, and the first k balls is in the group of first m balls, k > m.  = where A = “largest among the first k balls is in the group of first m balls drawn” B = “ (k+1)st ball has the largest number among all n balls”.  Since A and B are independent,  So, P (“selected ball has the largest number among all balls”)

Sharif University of Technology 30 Day-trading strategy: Solution  To maximize with respect to p, let:  So, p = e -1 ≈  This strategy can be used to “play the stock market”.  Suppose one gets into the market and decides to stay up to 100 days.  When to get out?  According to the solution, when the stock value exceeds the maximum among the first 37 days.  In that case the probability of hitting the top value over 100 days for the stock is also about 37%.

Sharif University of Technology 31 Example: Game of Craps  A pair of dice is rolled on every play, -Win: sum of the first throw is 7 or 11 -Lose: sum of the first throw is 2, 3, 12 -Any other throw is a “carry-over”  If the first throw is a carry-over, then the player throws the dice repeatedly until: -Win: throw the same carry-over again -Lose: throw 7 Find the probability of winning the game.

Sharif University of Technology 32 Game of Craps - Solution B : winning the game by throwing the carry-over C : throwing the carry-over P 1 : the probability of a win on the first throw Q 1 : the probability of loss on the first throw

Sharif University of Technology 33 Game of Craps - Solution For winning by carry-over, the probability of each throw (k) that does not count is: The probability that the player throws the carry-over k on the j-th throw is:

Sharif University of Technology 34 Game of Craps - Solution the probability of winning the game of craps is for the player. Thus the game is slightly advantageous to the house.