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2.2 Problems on minterm analysis  (Page 6/6)

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A computer store sells computers, monitors, printers. A customer enters the store. Let A , B , C be the respective events the customer buys a computer, a monitor, a printer. Assume the following probabilities:

  • The probability P ( A B ) of buying both a computer and a monitor is 0.49.
  • The probability P ( A B C c ) of buying both a computer and a monitor but not a printer is 0.17.
  • The probability P ( A C ) of buying both a computer and a printer is 0.45.
  • The probability P ( B C ) of buying both a monitor and a printer is 0.39
  • The probability P ( A C c ⋁ A c C ) of buying a computer or a printer, but not both is 0.50.
  • The probability P ( A B c ⋁ A c B ) of buying a computer or a monitor, but not both is 0.43.
  • The probability P ( B C c ⋁ B c C ) of buying a monitor or a printer, but not both is 0.43.

  1. What is the probability P ( A ) , P ( B ) , or P ( C ) of buying each?
  2. What is the probability of buying exactly two of the three items?
  3. What is the probability of buying at least two?
  4. What is the probability of buying all three?
% file npr02_19.m % Data for [link] % A = computer; B = monitor; C = printer minvec3DV = [A|Ac; A&B; A&B&Cc; A&C; B&C; (A&Cc)|(Ac&C); ... (A&Bc)|(Ac&B); (B&Cc)|(Bc&C)];DP = [1 0.49 0.17 0.45 0.39 0.50 0.43 0.43];TV = [A; B; C; (A&B&Cc)|(A&Bc&C)|(Ac&B&C); (A&B)|(A&C)|(B&C); A&B&C];disp('Call for mincalc') npr02_19Variables are A, B, C, Ac, Bc, Cc They may be renamed, if desired.Call for mincalc mincalcData vectors are linearly independent Computable target probabilities1.0000 0.8000 2.0000 0.61003.0000 0.6000 4.0000 0.37005.0000 0.6900 6.0000 0.3200The number of minterms is 8 The number of available minterms is 8Available minterm probabilities are in vector pma To view available minterm probabilities, call for PMA
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Data are P ( A ) = 0 . 232 , P ( B ) = 0 . 228 , P ( A B C ) = 0 . 045 , P ( A C ) = 0 . 062 ,

P ( A B ∪ A C ∪ B C ) = 0 . 197 and P ( B C ) = 2 P ( A C ) .

Determine P ( A ∪ B ∪ C ) and P ( A c B c C ) , if possible.

Repeat, with the additional data P ( C ) = 0 . 230 .

% file npr02_20.m % Data for [link] minvec3 DV = [A|Ac; A; B; A&B&C; A&C; (A&B)|(A&C)|(B&C); B&C - 2*(A&C)];DP = [ 1 0.232 0.228 0.045 0.062 0.197 0];TV = [A|B|C; Ac&Bc&Cc];disp('Call for mincalc') % Modification% DV = [DV; C];% DP = [DP 0.230 ]; npr02_20 Variables are A, B, C, Ac, Bc, Cc They may be renamed, if desired.mincalc Data vectors are linearly independentData probabilities are INCONSISTENT The number of minterms is 8The number of available minterms is 6 Available minterm probabilities are in vector pmaTo view available minterm probabilities, call for PMA disp(PMA)2.0000 0.0480 3.0000 -0.0450 % Negative minterm probabilities indicate4.0000 -0.0100 % inconsistency of data 5.0000 0.01706.0000 0.1800 7.0000 0.0450DV = [DV; C];DP = [DP 0.230];mincalc Data vectors are linearly independentData probabilities are INCONSISTENT The number of minterms is 8The number of available minterms is 8 Available minterm probabilities are in vector pmaTo view available minterm probabilities, call for PMA
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Data are: P ( A ) = 0 . 4 , P ( A B ) = 0 . 3 , P ( A B C ) = 0 . 25 , P ( C ) = 0 . 65 ,

P ( A c C c ) = 0 . 3 . Determine available minterm probabilities and the following,

if computable:

P ( A C c ∪ A c C ) , P ( A c B c ) , P ( A ∪ B ) , P ( A B c )

With only six items of data (including P ( Ω ) = P ( A ⋁ A c ) = 1 ), not all minterms are available. Try the additional data P ( A c B C c ) = 0 . 1 and P ( A c B c ) = 0 . 3 . Are these consistent and linearly independent? Are all minterm probabilities available?

% file npr02_21.m % Data for [link] minvec3 DV = [A|Ac; A; A&B; A&B&C; C; Ac&Cc];DP = [ 1 0.4 0.3 0.25 0.65 0.3 ];TV = [(A&Cc)|(Ac&C); Ac&Bc; A|B; A&Bc];disp('Call for mincalc') % Modification% DV = [DV; Ac&B&Cc; Ac&Bc];% DP = [DP 0.1 0.3 ]; npr02_21 Variables are A, B, C, Ac, Bc, Cc They may be renamed, if desired.Call for mincalc mincalcData vectors are linearly independent Computable target probabilities1.0000 0.3500 4.0000 0.1000The number of minterms is 8 The number of available minterms is 4Available minterm probabilities are in vector pma To view available minterm probabilities, call for PMADV = [DV; Ac&B&Cc; Ac&Bc];DP = [DP 0.1 0.3 ];mincalc Data vectors are linearly independentComputable target probabilities 1.0000 0.35002.0000 0.3000 3.0000 0.70004.0000 0.1000 The number of minterms is 8The number of available minterms is 8 Available minterm probabilities are in vector pmaTo view available minterm probabilities, call for PMA
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Repeat [link] with P ( A B ) changed from 0.3 to 0.5. What is the result? Explain the reason for this result.

% file npr02_22.m % Data for [link] minvec3 DV = [A|Ac; A; A&B; A&B&C; C; Ac&Cc];DP = [ 1 0.4 0.5 0.25 0.65 0.3 ];TV = [(A&Cc)|(Ac&C); Ac&Bc; A|B; A&Bc];disp('Call for mincalc') % Modification% DV = [DV; Ac&B&Cc; Ac&Bc];% DP = [DP 0.1 0.3 ]; npr02_22 Variables are A, B, C, Ac, Bc, Cc They may be renamed, if desired.Call for mincalc mincalcData vectors are linearly independent Data probabilities are INCONSISTENTThe number of minterms is 8 The number of available minterms is 4Available minterm probabilities are in vector pma To view available minterm probabilities, call for PMAdisp(PMA) 4.0000 -0.20005.0000 0.1000 6.0000 0.25007.0000 0.2500 DV = [DV; Ac&B&Cc; Ac&Bc];DP = [DP 0.1 0.3 ];mincalc Data vectors are linearly independentData probabilities are INCONSISTENT The number of minterms is 8The number of available minterms is 8 Available minterm probabilities are in vector pmaTo view available minterm probabilities, call for PMA disp(PMA)0 0.2000 1.0000 0.10002.0000 0.1000 3.0000 0.20004.0000 -0.2000 5.0000 0.10006.0000 0.2500 7.0000 0.2500
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Repeat [link] with the original data probability matrix, but with A B replaced by A C in the data vector matrix. What is the result? Does mincalc work in this case? Check results on a minterm map.

% file npr02_23.m % Data for [link] minvec3 DV = [A|Ac; A; A&C; A&B&C; C; Ac&Cc];DP = [ 1 0.4 0.3 0.25 0.65 0.3 ];TV = [(A&Cc)|(Ac&C); Ac&Bc; A|B; A&Bc];disp('Call for mincalc') % Modification% DV = [DV; Ac&B&Cc; Ac&Bc];% DP = [DP 0.1 0.3 ];npr02_23 Variables are A, B, C, Ac, Bc, CcThey may be renamed, if desired. Call for mincalcmincalc Data vectors are NOT linearly independentWarning: Rank deficient, rank = 5 tol = 5.0243e-15 Computable target probabilities1.0000 0.4500 The number of minterms is 8The number of available minterms is 2 Available minterm probabilities are in vector pmaTo view available minterm probabilities, call for PMA DV = [DV; Ac&B&Cc; Ac&Bc];DP = [DP 0.1 0.3 ];mincalc Data vectors are NOT linearly independentWarning: Matrix is singular to working precision. Computable target probabilities1 Inf % Note that p(4) and p(7) are given in data 2 Inf3 Inf The number of minterms is 8The number of available minterms is 6 Available minterm probabilities are in vector pmaTo view available minterm probabilities, call for PMA
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Source:  OpenStax, Applied probability. OpenStax CNX. Aug 31, 2009 Download for free at http://cnx.org/content/col10708/1.6
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