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Our lesson

Our lesson. Experimental Probability. Warm Up. Find the value of each expression P(6,3) P(9,8) 9 ! 9 . 8 . 7 . 6 P(5,2). Let Us Review. The symbol n P r is used to indicate the number of permutations n objects taken r at a time.

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Our lesson

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  1. Our lesson Experimental Probability

  2. Warm Up • Find the value of each expression • P(6,3) • P(9,8) • 9! • 9 . 8 . 7 . 6 • P(5,2)

  3. Let Us Review • The symbol nPr is used to indicate the number of permutations n objects taken r at a time. nPr = n!/ (n - r)! • The numbers of n objects is n!. • If a set of n elements has n1 elements of one kind, n2 of another kind alike and so on, then the number of permutations , P, of the n elements taken n at a time is given by: P = n! n1!.n2!.......

  4. Probability – Introduction • Probability is a numerical measure of the likelihood of occurrence of an event. Probability mathematically lies between two limits 0 and 1. • 0 represents ‘Impossibility’, 1 (Something simply cannot happen • 1 represents ‘Certainity’ (guaranteed to happen) • 0 and 1 are the two extremes. Most probabilities lie somewhere between them. • Probability is the ratio of the different number of ways a trial can succeed (or fail) to the total number of ways in which it may result.

  5. Probability – Basics • An experiment is a method by which observations are made. • Example:The act of rolling a fair die, flipping an honest coin • A possible result of a probability experiment is called an outcome. • Example: Head or Tail (Experiment-Tossing a fair coin) • 1 or 2 0r 3 or 4 or 5 or 6 (Experiment - Rolling a dice) • The sample space is the set of all possible outcomes for a given experiment. Each possible result of such a study is represented by one and only one point in the sample space, which is usually denoted by S • Example: Experiment - Rolling a number cube once • Sample space S = {1,2,3,4,5,6}

  6. Probability - Event An event(E) is a one or more outcome of an experiment. Any event which consists of a single outcome in the sample space is called an elementary or simple event. Events which consist of more than one outcome are called compound events. Example: Simple event -Tossing a fair coin once Sample space, S ={ H, T} Event A = ‘ Head showing up’ = {H} Event B = ‘Tail showing up’ = {T}

  7. Probability - Event Example: Compound events -Tossing 2 fair coins Sample Space, S = {HH, HT, TH, TT} Event A = ‘Two heads showing up’ = {HH} Event B = ‘One head and one tail’ = {HT, TH} A set of outcomes of an event are said to be equally likely if they all have the same choice of happening.

  8. Experimental Probability Experimental Probability or Estimated Probability is the chances of something happening, based on repeated testing and observing results. It is the ratio of the number of times an event occurred to the number of times tested. To find the experimental probability of winning a game, one must play the game many times, then divide the number of games won by the total number of games played.

  9. The frequencyof the event fr (E) is the number of times the event E occurs.The number of times that the experiment is performed is called the number of trials or the sample size (N) The relative frequency or experimental probability of the event E is the fraction of times E occurs. Now, Experimental Probability of event (E) is given by, P(E) = fraction of times E occurs = fr (E) / N

  10. Probability of Success If a trial must result in any of n equally likely ways, and if s is the number of successful ways and f is the number of failing ways, the probability of success (p) is P = s / (s + f) , where s + f = n

  11. Example: What is the probability that a die will land with a 3 showing on the upper face? A die has a total of 6 sides. Therefore the die can land with a 3 face up 1 favorable way and 5 unfavorable ways. so, s = 1 and f = 5, P = ? P = 1/ (1+5) = 1/6

  12. Probability of Failure If a trial results in any of n equally likely ways, and s is the number of successful ways and f is the number of failures, the probability of failure (q) is q = f / (s + f) , where s + f = n or q = n-s / n In any event, the probability of Success plus the probability of failure is equal to 1. So, p + q = s/(s+f) + f/(s+f) p + q = 1 or q = P - 1

  13. Probability of Failure - example Example: What is the probability of not drawing a black marble from a box containing 6 white, 3 red, and 2 black marbles. Number of black marbles = 2. So, s = 2 Number of marbles apart from black = 9. So, f = 9 The probability of drawing a black marble(p) from the box is , P = s / (s + f) = 2 / (2 + 9) = 2/11 Since p + q = 1, The probability of not drawing a black marble(q) is, q = 1 - p = 1 – 2/11 = 9/11

  14. Relative Frequency • Relative frequency is another term for proportion; it is the value calculated by dividing the number of times an event occurs by the total number of times an experiment is carried out. • If an experiment is repeated n times, and event E occurs r times, then the relative frequency of the event E is defined to be, • Relative Frequency of an event, rfn(E) = r/n • Example: Experiment-Tossing a fair coin 30 times (n = 30) • Event E = ‘Tails' • Result: 20 heads, 10 tails, so r = 10 • Relative frequency: rfn(E) = r/n = 10/30 = 1/3 = 0.33

  15. i.e P(E) = lim rfn(E) n  Limiting value of the relative frequency If an experiment is repeated many, many times without changing the experimental conditions, the relative frequency of any particular event will settle down to some value. The probability of the event can be defined as the limiting value of the relative frequency.

  16. For example, if the experiment of Tossing a fair coin is repeated many more times , the relative frequency of the event ‘tails' will settle down to a value of approximately 0.5 which is equal to the probability of tossing a coin once. So, probability has two approaches. One is theoretical and the other is experimental .Theoretical probability is the ratio of the number of ways the event can occur to the total number of possibilities in the sample space. P(E) = number of ways to get what you want/Total number of possibilities P(E) = n(E) / n(S)

  17. Experimental Probability-Properties Some Properties of Experimental Probability : Let S = {S1, S2, S3…Sn} be a sample space and let P(Si) be the estimated probability of the event {Si }. Then • The experimental probability of each outcome is a number between 0 and 1. i.e0 <= P(Si) <= 1 • The experimental probabilities of all the outcomes add up to 1. i.e P(S1) + P(S2) + ... + P(Sn) = 1 c) The experimental probability of an event E is the sum of the estimated probabilities of the individual outcomes in E If E = {e1, e2, e3, …………er}, then P(E) = P(e1) + P(e2) + ... + P(er).

  18. Experimental Probability –Example 1 Example 1: Toss a coin 20 times. What is the probability of head showing up 11 times? The frequency of the event that heads comes up is fr(E) = 11. The experiment is performed 20 times. So the sample size is 20 The relative frequency or the experimental probability of the event P(Head showing up) 11 times is P(E) = fr(E) / N = 11/20

  19. Experimental Probability –Example 2 Example 2: In a dart game, a player hits the bull's eye 6 times out of 25 trials. What is the statistical probability that he will hit the bull's eye on the next throw ? Here, the frequency of the event, player hitting the bull’s eye is fr(E) = 6 The number of trials, N = 25. The experimental probability of the player hitting the bulls eyes is, P(E) = fr(E) / N = 6/25

  20. Probability Distribution The collection of the estimated probabilities of all the outcomes is the experimental probability distribution or estimated probability distributionorrelative frequency distribution Example: If 10 rolls of a die resulted in the outcomes 2, 1, 4, 4, 5, 6, 1, 2, 2, 1, then the associated estimated probability distribution is the one shown in the following table.

  21. Numerical Expectation Expectation is the average of the values you would get in conducting an experiment or trial, exactly the same way many times. There are two types,one is a numerical expectation and the other is a mathematical expectation. Numerical Expectation: If the probability of success in one trial is p, and k is the total number of trials, then kp is the expected number of successes (En) in the k trials. i.e. En = kp

  22. Example: Suppose you toss a coin 30 times. What would be your numerical expectation of heads showing up? The probability of head showing up in one trial is ½ The number of trials, k= 30 The numerical expectation of heads showing up in 30 trials is, En = kp = 30(1/2) = 15 In words, you would expect (on the average) to win heads 15 times in 30 tosses.

  23. Probability Distribution The collection of the estimated probabilities of all the outcomes is the experimental probability distribution or estimated probability distributionorrelative frequency distribution Example: If 10 rolls of a die resulted in the outcomes 2, 1, 4, 4, 5, 6, 1, 2, 2, 1, then the associated estimated probability distribution is the one shown in the following table

  24. Mathematical Expectation Mathematical Expectation: If, in the event of a successful result, amount a, is to be received and the probability of success of that event is p, then ap is the mathematical expectation (Em). i.e Em = ap a = amount you stand to win, P = probability of success Example: If you were to buy 1 of 500 raffle tickets for a video recorder worth $325.00, what would be your mathematical expectation?

  25. The probability of buying 1 out of 500 tickets is = 1/500 The video recorder of the amount to be received = $325 The product of the amount you stand to win and the probability of winning is, Em = ap = 325 * (1/500) = $ 0.65 So, you would not want to pay more than 65 cents for the ticket, unless, of course the raffle were for a worthy cause.

  26. Your Turn • A pair of dice (one red, one green) is cast 40 times, and on 8 of these occasions, the sum of the numbers facing up is 7. What is the experimental probability of the outcome 7? • 2. There are 2 entry doors and 3 staircases in your school.  How many ways are there to enter the building and go to the second floor?  List the sample space. • 3. A number cube is tossed 20 times and lands on 2 three times and on 4 four times. Find the experimental probability of not landing on 2.

  27. Your Turn • I flipped a coin 20 times and the results are below. What is the probability of getting tails? H H H T H T T H T H T T T T H T H H T T 5. Suppose you roll a 6 sided die 25 times and the results are below. What is the probability of getting a 4?6 4 2 1 2 3 4 6 5 4 2 2 1 5 3 4 6 2 1 2 1 3 5 4 4 6. There is 3.1% chance that a baby boy will be born. What is the chance that a baby girl will be born?

  28. Your Turn 7. Among eight helicopters sent to rescue American hostages in Iran, three helicopters failed to operate properly. Given the same conditions, what is the probability of failure for a helicopter? 8. If a die is rolled, what is the probability of an odd number not showing on the upper face?

  29. Your Turn 9. A man has 3 nickels, 2 dimes, and 4 quarters in his pocket. If he draws a single coin from his pocket, what is the probability that . he will draw a half-dollar? ? 10. A box contains 6 hard lead pencils and 12 soft lead pencils. What is the probability of drawing a soft lead pencil from the box ?

  30. Refreshment time

  31. Click on the ball to play a game

  32. In 1993, there were approximately 10,000 fast food outlets in the US that specialized in Mexican food. Of these, the largest were Taco Bell with 4,809 outlets, Taco John's with 430 outlets and Del Taco with 275 outlets. Find the experimental probability that a fast food outlet that specializes in Mexican food is none of the above

  33. A box contains a few black and red tiles. John picked up two tiles from the box at a time and repeated this 5 times. The results of this are recorded as shown in the table. Find the experimental probability that John will pick up two black tiles from the box next time.

  34. 3. Bill placed the cards labeled 1 to 10 in a bag. He drew one card from the bag 45 times. If his draws included ten 2's, three 4's, and two 6's, then what is the experimental probability of drawing a 2, 4, or 6?

  35. Let’s summarize what we have learnt today • Probability is the ratio of the different number of ways a trial can succeed (or fail) to the total number of ways in which it may result • Probability mathematically lies between two limits 0 and 1 • An experiment is a method by which observations are made. • A possible result of a probability experiment is called an outcome. • The sample space is the set of all possible outcomes for a given experiment. Each possible result of such a study is represented by one and only one point in the sample space, which is usually denoted by S

  36. Let’s summarize what we have learnt today 6. An event(E) is a one or more outcome of an experiment. 7. A set of outcomes of an event are said to be equally likely if they all have the same choice of happening. 8. The probability of success (p) isP = s / (s + f) 9. The probability of failure (q) is , q = f / (s + f) 10. Relative frequency is the value calculated by dividing the number of times an event occurs by the total number of times an experiment is carried out. 11. The probability of the event is defined as the limiting value of the relative frequency.

  37. Let’s summarize what we have learnt today 12. Experimental Probability or Estimated Probability is the ratio of the number of times an event occurred to the number of times tested. P = fr(E)/N 13. The numerical expectation of an event is the product of the probability of success in one trial and the total number of trials. En = kp 14. The mathematical expectation of an event is the product of amount to be received and the probability of success of that event.Em = ap

  38. You did a great job ! Set your goals high, and don't stop till you get there

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