#$&* course Mth 152 November 28 - 5:19pm Question: `q001. Note that there are 8 questions in this assignment.
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Given Solution: You obtained four 5's, which total 4 * 5 = 20. You obtained seven 6's, which total 7 * 6 = 42. You obtained nine 7's, which total 9 * 7 = 63. You obtained six 8's, which total 6 * 8 = 48. You obtained three 9's, which total 3 * 9 = 27. The total of all the outcomes is therefore 20 + 42 + 63 + 48 + 27 = 200. Since there are 4 + 7 + 9 + 6 + 3 = 29 outcomes (i.e., four outcomes of 5 plus 7 outcomes of 6, etc.), the mean is therefore 200/29 = 6.7, approximately. This series of calculations can be summarized in a table as follows: Result Frequency Result * frequency 5 4 20 6 7 42 7 9 63 8 6 48 9 3 27 9 3 27 ___ ____ ____ 29 200 mean = 200 / 29 = 6.7 Self-critique: OKAY ------------------------------------------------ Self-critique rating: OKAY ********************************************* Question: `q002. The preceding problem could have been expressed in the following table: Total Number of Occurrences 5 4 6 7 7 9 8 6 9 3 This table is called a frequency distribution. It expresses each possible result and the number of times each occurs. You found the mean 6.7 of this frequency distribution in the preceding problem. Now find the standard deviation of the distribution. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: We are going to make another table to lie out all the information together. We have the results, the frequency, the results of the frequency (multiplication of the two), the deviation from the freq. distribution 6.7 (dev of sq), the deviation squared (dev sp), and the final deviation frequency which is the frequency multiplied by the dev sq (final freq.) Results Freq. Result of freq/ Dev. Of Sq/ Dev sq/ final freq. 5 4 20 1.7 2.89 11.56 6 7 42 .7 0.49 3.4 7 9 63 0.3 0.09 0.6 8 6 48 1.3 1.69 10.2 9 3 27 2.3 5.29 15.9 ___________________________________________ 29 200 41.7 confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: We must calculate the square root of the 'average' of the squared deviation. We calculate the deviation of each result from the mean, then find the squared deviation. To find the total of the squared deviations we must add each squared deviation the number of times which is equal to the number of times the corresponding result occurs. For example, the first result is 5 and it occurs four times. Since the deviation of 5 from the mean 6.7 is 1.7, the squared deviation is 1.7^2 = 2.89. Since 5 occurs four times, the squared deviation 2.89 occurs four times, contributing 4 * 2.89 = 11.6 to the total of the squared deviations. Using a table in the manner of the preceding exercise we obtain Result Freq Result * freq Dev Sq Dev Sq Dev * freq 5 4 20 1.7 2.89 11.6 6 7 42 .7 0.49 3.4 7 9 63 0.3 0.09 0.6 8 6 48 1.3 1.69 10.2 9 3 27 2.3 5.29 15.9 ___ ____ ____ ___ 29 200 41.7 mean = 200 / 29 = 6.7 'ave' squared deviation = 41.7 / (29 - 1) = 1.49 std dev = `sqrt(1.49) = 1.22 Self-critique: Okay ------------------------------------------------ Self-critique rating: Okay ********************************************* Question: `q003. If four coins are flipped, the possible numbers of 'heads' are 0, 1, 2, 3, 4. Suppose that in an experiment we obtain the following frequency distribution: # Heads Number of Occurrences 0 4 1 20 2 22 3 13 4 3 What is the mean number of 'heads' and what is the standard deviation of the number of heads from this mean? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The mean equals the results divided by the frequency = 115 /62 = 1.86 Heads/Result Freq. Result (Dev Sq.) Squared Dev 0 4 0 1.89 3.5 14 1 20 20 0.86 0.7 14 2 22 44 0.14 0.1 2 3 13 39 1.14 1.3 17 4 3 12 2.14 4.6 14 _________________________________________________ 10 62 115 6.17 10.2 61 To find the squared deviation we take the frequency divide it by the result. 61/62 = .98 which when squared is .989 or .99 confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: Using the table in the manner of the preceding problem we obtain the following: Result Freq Result * freq Dev Sq Dev Sq Dev * freq 0 4 0 1.86 3.5 14 1 20 20 0.86 0.7 14 2 22 44 0.14 0.1 2 3 13 39 1.14 1.3 17 4 3 12 2.14 4.6 14 ___ ____ ____ ___ 62 115 61 mean = 115 / 62 = 1.86 approx. Note that the mean must be calculated before the Dev column is filled in. 'ave' squared deviation = 61 / 62 = .98 std dev = `sqrt(.98) = .99 Self-critique: okay ------------------------------------------------ Self-critique rating: okay ********************************************* Question: `q004. If we rolled 2 dice 36 times we would expect the following distribution of totals: Total Number of Occurrences 2 1 3 2 4 3 5 4 6 5 7 6 8 5 9 4 10 3 11 2 12 1 What is the mean of this distribution and what is the standard deviation? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The mean equals the results divided by the frequency = 252/36=7 Results Frequency result dev. Sq dev. Sq dev. *freq 2 1 2 5 25 25 3 2 6 4 16 32 4 3 12 3 9 27 5 4 20 2 4 16 6 5 30 1 1 5 7 6 42 0 0 0 8 5 40 1 1 5 9 4 36 2 4 16 10 3 30 3 9 27 11 2 22 4 16 32 12 1 12 5 25 25 ______________________________________________________ 77 36 252 30 110 210 To find the squared deviation we take the sq, dev frequency and divide it by the result: 210/36=5.8. The standard deviation squared is 2.4 confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: Using the table in the manner of the preceding problem we obtain the following: The mean Result Freq Result * freq Dev Sq Dev Sq Dev * freq 2 1 2 5 25 25 3 2 6 4 16 32 4 3 12 3 9 27 5 4 20 2 4 16 6 5 30 1 1 5 7 6 42 0 0 0 8 5 40 1 1 5 9 4 36 2 4 16 10 3 30 3 9 27 11 2 22 4 16 32 12 1 12 5 25 25 ___ ____ ____ ___ 36 252 210 mean = 252 / 36 = 7. Note that the mean must be calculated before the Dev column is filled in. 'ave' squared deviation = 210 / 36 = 5.8 approx. std dev = `sqrt(5.8) = 2.4 approx. Self-critique: okay ------------------------------------------------ Self-critique rating: okay ********************************************* Question: `q005. If we flip n coins, there are C(n, r) ways in which we can get r 'heads' and 2^n possible outcomes. The probability of r 'heads' is therefore C(n, r) / 2^n. If we flip five coins, what is the probability of 0 'heads', of 1 'head', of 2 'heads', of 3 'heads', of 4 'heads', and of 5 'heads'? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: If we flip 5 coins, then n = 5. To get 0 'heads' we find C(n, r) with n = 5 and r = 0, obtaining C(5,0) = 1 way to get 0 'heads' out of the 2^5 = 32 possibilities for a probability of 1 / 32. To get 1 'heads' we find C(n, r) with n = 5 and r = 1, obtaining C(5,1) = 5 ways to get 1 'heads' out of the 2^5 = 32 possibilities for a probability of 5 / 32. To get 2 'heads' we find C(n, r) with n = 5 and r = 2, obtaining C(5,2) = 10 ways to get 2 'heads' out of the 2^5 = 32 possibilities for a probability of 10 / 32. To get 3 'heads' we find C(n, r) with n = 5 and r = 3, obtaining C(5,3) = 10 ways to get 3 'heads' out of the 2^5 = 32 possibilities for a probability of 10 / 32. To get 4 'heads' we find C(n, r) with n = 5 and r = 4, obtaining C(5,4) = 5 ways to get 4 'heads' out of the 2^5 = 32 possibilities for a probability of 5 / 32. To get 5 'heads' we find C(n, r) with n = 5 and r = 5, obtaining C(5,5) = 1 way to get 5 'heads' out of the 2^5 = 32 possibilities for a probability of 1 / 32. Self-critique: ------------------------------------------------ Self-critique rating: ********************************************* Question: `q006. The preceding problem yielded probabilities 1/32, 5/32, 10/32, 10/32, 5/32 and 1/32. On 5 flips, then, we the expected values of the different numbers of 'heads' would give us the following distribution: : # Heads Number of Occurrences 0 1 1 5 2 10 3 10 4 5 5 1 Find the mean and standard deviation of this distribution. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: Using the table in the manner of the preceding problem we obtain the following: Result Freq Result * freq Dev Sq Dev Sq Dev * freq 0 1 0 2.5 6.25 6.25 1 5 5 1.5 2.25 11.25 2 10 20 0.5 0.25 2.50 3 10 30 0.5 0.25 2.50 4 5 20 1.5 2.25 12.25 5 1 5 2.5 6.25 6.25 ___ ____ ____ ___ 32 80 40.00 mean = 80 / 32 = 2.5. Note that the mean must be calculated before the Dev column is filled in. 'ave' squared deviation = 40 / 32 = 1.25. Thus std dev = `sqrt(1.25) = 1.12 approx. Self-critique: ------------------------------------------------ Self-critique rating: ********************************************* Question: `q007. Suppose that p is the probability of success and q the probability of failure on a coin flip, on a roll of a die, or on any other action that must either succeed or fail. If a trial consists of that action repeated n times, then the average number of successes on a large number of trials is expected to be n * p. For large values of n, the standard deviation of the number of successes is expected to be very close to `sqrt( n * p * q ). For values of n which are small but not too small, the standard deviation will still be close to this number but not as close as for large n. If the action is a coin flip and 'success' is defined as 'heads', then what is the value of p and what is the value of q? For this interpretation in terms of coin flips, if n = 5 then what is n * p and what does it mean to say that the average number of successes will be n * p? In terms of the same interpretation, what is the value of `sqrt(n * p * q) and what does it mean to say that the standard deviation of the number of successes will be `sqrt( n * p * q)? How does this result compare with the result you obtained on the preceding problem? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: We first identify the quantities p and q for a coin flip. Success is 'heads', which for a fair coin occurs with probability .5. Failure therefore has probability 1 - .5 = .5. Now if n = 5, n * p = 5 * .5 = 2.5, which represents the mean number of 'heads' on 5 flips. The idea that the mean number of occurrences of some outcome with probability p in n repetitions is n * p should by now be familiar (e.g., from basic probability and from the idea of expected values). For n = 5, we have `sqrt(n * p * q) = `sqrt(5 * .5 * .5) = `sqrt(1.25) = 1.12, approx.. In the preceding problem we found that the standard deviation expected on five flips of a coin should be exactly 1. This differs from the estimate `sqrt(n * p * q) by a little over 10%, which is a fairly small difference. STUDENT QUESTION Solution above—could you explain the q? INSTRUCTOR RESPONSE p is the probability of success q is the probability of failure p + q = 1 so q = 1 - p. Self-critique: ------------------------------------------------ Self-critique rating: ********************************************* Question: `q008. Suppose that p is the probability of success and q the probability of failure on a coin flip, on a roll of a die, or on any other action that must either succeed or fail. If a trial consists of that action repeated n times, then the average number of successes on a large number of trials is expected to be n * p. The standard deviation of the number of successes is expected to be `sqrt( n * p * q ). If the action is a roll of a single die and a success is defined as rolling a 6, then what is the probability of a success and what is the probability of a failure? If n = 12 that means that we count the number of 6's rolled in 12 consecutive rolls of the die, or alternatively that we count the number of 6's when 12 dice are rolled. How many 6's do we expect to roll on an average roll of 12 dice? What do we expect is the standard deviation the number of 6's on rolls of 12 dice? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: P = is the probability of success q = the probability of failure on a coin flip, on a roll of a die, or on any other action that must either succeed or fail n= times of action n * p. = average number of successes on a large number of trials is expected to be `sqrt( n * p * q ) = standard deviation of the number of successes is expected to be In this problem n = 12. Each die has 6 sides and chances. We were asked to find the probability of successes and failures. A successful role of the die would be 6 so in terms of success the probability of rolling a 6 would be 1/6. Failure then would be a 5/6 probable chance. Next we set up the problem. N=12. P=1/6. Q=5/6. N*P=12*1/6 Sqrt ( 12 * 1/6 * 5/6 ) = sqrt (3.5 * 0.4 * 0.91) = 1.27 or 1.3 confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: We first identify the quantities p and q for a rolling a die. Success is defined in this problem as getting a 6, which for a fair die occurs with probability 1/6. Failure therefore has probability 1 - 1/6 = 5/6. Now if n = 12, n * p = 12 * 1/6 = 2, which represents the mean number of 6's expected on 12 rolls. This is the result we would expect. For n = 12, we have `sqrt(n * p * q) = `sqrt(12 * 1/6 * 5/6) = `sqrt(1.66) = 1.3, approx.. Self-critique: okay ------------------------------------------------ Self-critique rating: okay " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: #*&!