Subjective, Experimental, and Theoretical Probability - Assignment Example

Assignment 3-2 Project II (Teresa Phares) Math215 Table of Content Introduction 2 Part ive Probability 3 Part 2 Empirical (or Experimental) Probability 5 Part 3 Theoretical Probability 6 Conclusion 10 References: 11 Appendix: 12 Introduction Probability is the mathematics of chance. The concepts of probability are as old as humans. Tossing a coin and getting head, picking a card from deck, rolling a die and getting a certain number are some of the examples where probability concepts are applied. This paper (project) will discuss three types of probability subjective, empirical, and classical (theoretical) using an experiment ‘Toss of Coins’. The experiment ‘Toss of Coins’ will consist of following steps: 1. Take ten pennies 2. Put pennies in a large cup 3. Shake cup to ensure mixing 4. Toss the pennies on a flat counter 5. Record the number of heads 6. Repeat steps 1 to 5 for 60 tosses. Finally, subjective (guess) and theoretical, and empirical (experiment) and theoretical probabilities will be compared using double bar graphs. Part 1 Subjective Probability Subjective probability is made by a person’s knowledge of the situation. It is an educated guess as to the chances of an event occurring. This guess is based on the person’s experience and evaluation of a solution. The subjective approach to probability is needed when there is no repeatable random experiment (Doane & Seward 2007). Tossing 10 coins only once have eleven possible outcomes, they are zero heads, one head, two heads….or ten heads. These eleven outcomes are mutually exclusive and all inclusive. Some of the characteristics of probability distribution are The sum of all of the probabilities must be 1.000 (). Each probability is between zero and one, inclusive (). 1. What percent of the time do you expect to get 5 heads? 24% 2. What percent of the time do you expect to get 3 heads? 12% 3. What percent of the time do you expect to get 7 heads? 12% 4. What percent of the time do you expect to get no heads? 0.1% 5. What percent of the time do you expect to get all heads? 0.1% 6. Use your guesses above to fill in the chart below for all of the outcomes. Table 1: Subjective Probability (My Guess) X = Heads 0 1 2 3 4 5 6 7 8 9 10 Prob(X) 0.001 0.010 0.040 0.120 0.200 0.240 0.200 0.120 0.490 0.010 0.001 7. Adjust your guesses for 1-6 above until the two rules of probability are satisfied (Each value is between 0 and 1 and the sum of all of the probabilities is 1. 8. After throwing the 600 coins, how many heads would you expect? 300 9. How is the probability of getting three heads (# 2 above) related to the probability of getting three tails (# 3 above) ? Explain your answer. The probability of getting three heads is equal to probability of getting three tails. For every single toss of coin, one can get either head or tail. When one will toss 10 coins simultaneously and get three heads on three coins that means he/she will also get seven tails on rest of the coins. Similarly, if one will get three tails on three coins that means he/she will also get seven heads on rest of the coins. Both types of experiment are same (getting three heads or three tails) as probability of getting head or tail on a single toss is equal (0.5), therefore, one will get same result. 10. How is the probability of getting three heads related to the probability of getting seven tails? Explain your answer. When one will toss 10 coins only once and gets three heads on three coins that means he/she will also get seven tails on rest of the coins because for any coin toss, there is only two possible outcomes either head or tail (mutually exclusive events). For example, toss of two coins will have following outcomes: HH, HT, TH, and TT. If on a single trial one will get one head that means he/she will also get one tail (i.e. HT or TH). Part 2 Empirical (or Experimental) Probability Probabilities can be computed for situations that do not use sample spaces. In such cases, frequency distributions are used and the probability is called empirical probability. Empirical probability uses frequency distributions, and it is defined as the frequency of an event divided by the total number of frequencies (Bluman 2005): Figure 1: Cumulative Percent Heads vs. Toss Number Figure 1 shows the cumulative percent heads of toss of coins experiment. From figure 1, it can be seen that as the number of toss for coins increases the cumulative percent heads approaches to 0.5. This is because for any toss, probability of getting head or tail is 0.5. Therefore, as one will increase number of tosses, the overall result (cumulative percent) for heads or tails will be close to 0.5. In ‘Toss of Coins’ experiment, 290 heads on 600 tosses of coins (Appendix: table 3) came, which is equal to 48.33% that is very near to 50%. Part 3 Theoretical Probability Classical Probability Sample spaces are used in classical probability to determine the numerical probability that an event will occur. Classical probability is defined as the number of ways (outcomes) the event can occur divided by the total number of outcomes in the sample space. The formula for determining the probability of an event E is Binomial Distribution A probability distribution consists of the values of a random variable and their corresponding probabilities. The binomial distribution is one of special probability distribution that has many uses, such as in gambling, in inspecting parts, and in other areas. A binomial distribution is obtained from a probability experiment called a binomial experiment. The experiment must satisfy these conditions (Bluman 2005): 1) Each trial can have only two outcomes or outcomes that can be reduced to two outcomes. The outcomes are usually considered as a success or a failure. 2) There is a fixed number of trials. 3) The outcomes of each trial are independent of each other. 4) The probability of a success must remain the same for each trial. In order to determine the probability of a success for a single trial of a probability experiment, the following formula is used. Where, n = the total number of trials, x = the number of successes (1, 2, 3. . . n) and p = the probability of a success. The formula has three parts: determines the number of ways a success can occur. is the probability of getting x successes, and is the probability of getting (n-x) failures. The experiment ‘Toss of Coins’ is an example of binomial distribution. Tossing 10 coins (once) fits all of the conditions of a binomial distribution 1) Two outcomes are possible on each trial. Heads is a success. 2) There are 10 identical trials (each coin is one trial) so N = 10 3) Each coin is independent of the others. 4) The probability of success is the same on each coin so p = 0.5 Table 3, summarizes the results of subjective (guess), empirical (experiment), and theoretical probabilities. Table 2: Subjective , Empirical, and Theoretical Probability X = No of Successes Subjective Probability of Success Actual Number of Times Empirical Probability Theoretical Probability 0 0.001 0 0.0000 0.000976563 1 0.010 1 0.0167 0.009765625 2 0.049 1 0.0167 0.043945313 3 0.120 9 0.1500 0.1171875 4 0.200 16 0.2667 0.205078125 5 0.240 15 0.2500 0.24609375 6 0.200 10 0.1667 0.205078125 7 0.120 3 0.0500 0.1171875 8 0.049 5 0.0833 0.043945313 9 0.010 0 0.0000 0.009765625 10 0.001 0 0.0000 0.000976563 Sums 1.000 60 1 1 Figure 2: Subjective Vs Theoretical Probability of X Successes Figure 2 shows the double bar graph of the “Theoretical Probability of X Successes” and the subjective probabilities that were guessed in the table 1 in Part One. It is merely a chance (again probability) that both bars are same. The bar graph of the subjective probabilities will be different for different people as guesses for X successes (0, 1, 2 …10 heads) will be not same for all people. However, the bar graph of “Theoretical Probability of X Successes” will be same. Figure 3: Empirical Vs Theoretical Probability of X Successes Figure 3 shows the double bar graph of the “Theoretical Probability of X Successes” and the “Empirical Probability of Success.” There is slight difference in bars for both the “Theoretical Probability of X Successes” and the “Empirical Probability of Success.” The bar graph of the “Empirical Probability of Success” will be different for different experiment. However, as the experiment will get bigger and bigger, the bar graph of the “Empirical Probability of Success” will tend similar to bar graph of the “Theoretical Probability of X Successes”. Conclusion In conclusion, for a large experiment, the results of the “Empirical Probability of Success” and the “Theoretical Probability of X Successes” will be more or less same. However, in case of the “Subjective Probability of X Successes”, it will be different based on person’s knowledge of the situation. References: Bluman, A.G. (2005). Probability Demystified. McGraw-Hill: New York Doane D.P. & Seward L.E. (2007). Applied Statistics in Business and Economics. McGraw-Hill/Irwin: New York Appendix: Table 3: Coin Toss Results Toss Number Number of Heads Cumulative Heads Total Coins Tossed Cumulative Percent Heads 1 3 3 10 0.3000 2 5 8 20 0.4000 3 3 11 30 0.3667 4 8 19 40 0.4750 5 4 23 50 0.4600 6 3 26 60 0.4333 7 2 28 70 0.4000 8 5 33 80 0.4125 9 4 37 90 0.4111 10 4 41 100 0.4100 11 3 44 110 0.4000 12 4 48 120 0.4000 13 6 54 130 0.4154 14 4 58 140 0.4143 15 3 61 150 0.4067 16 5 66 160 0.4125 17 4 70 170 0.4118 18 4 74 180 0.4111 19 4 78 190 0.4105 20 4 82 200 0.4100 21 8 90 210 0.4286 22 4 94 220 0.4273 23 4 98 230 0.4261 24 5 103 240 0.4292 25 5 108 250 0.4320 26 4 112 260 0.4308 27 7 119 270 0.4407 28 3 122 280 0.4357 29 6 128 290 0.4414 30 5 133 300 0.4433 31 3 136 310 0.4387 32 8 144 320 0.4500 33 4 148 330 0.4485 34 8 156 340 0.4588 35 5 161 350 0.4600 36 3 164 360 0.4556 37 5 169 370 0.4568 38 6 175 380 0.4605 39 5 180 390 0.4615 40 6 186 400 0.4650 41 8 194 410 0.4732 42 5 199 420 0.4738 43 4 203 430 0.4721 44 5 208 440 0.4727 45 5 213 450 0.4733 46 3 216 460 0.4696 47 4 220 470 0.4681 48 6 226 480 0.4708 49 1 227 490 0.4633 50 6 233 500 0.4660 51 6 239 510 0.4686 52 7 246 520 0.4731 53 6 252 530 0.4755 54 5 257 540 0.4759 55 6 263 550 0.4782 56 5 268 560 0.4786 57 5 273 570 0.4789 58 4 277 580 0.4776 59 7 284 590 0.4814 60 6 290 600 0.4833 Read More