(%i1) "Cornelsen Lk, S. 115, Nr. 1"$

(%i2) "a)"$

(%i3) "Anzahl der Pfade = 2^3 = 8"$

(%i5) "b) Anzahl der Pfade zu Kasten Nr. 2"$

(%i6) "3 ueber 2"$

(%i7) "Ziehen ohne Zurücklegen, ungeordnet"$

(%i8) binomial(3,2);
(%o8)                                  3

(%i9) "c)"$

(%i10) "B(n;p;k) mit n=3, p=0,5 und 0<=k<=3"$

(%i11) "Parameter der Binomialverteilung"$

(%i12) bin_n : 3$

(%i13) bin_p : 0.5$

(%i14) bin_q : 1-bin_p;
(%o14)                                0.5

(%i15) "Erwartungswert ew"$

(%i16) bin_ew : bin_n * bin_p;
(%o16)                                1.5

(%i17) "Standardabweichung sigma"$

(%i18) bin_sigma : ev(sqrt(bin_n * bin_p * bin_q),numer);
(%o18)                         0.86602540378444

(%i20) "P(ew-sigma <= X <= ew+sigma) (1-Sigma-Umgebung)"$

(%i21) sumBin(bin_n,ceiling(bin_ew-bin_sigma), floor(bin_ew+bin_sigma), bin_p)$
P( 1 <= X <= 2 ) =  0.75 

(%i22) "P(ew-2*sigma <= X <= ew+2*sigma) (2-Sigma-Umgebung)"$

(%i23) sumBin(bin_n,ceiling(bin_ew-2*bin_sigma), floor(bin_ew+2*bin_sigma), bin_p)$
P( 0 <= X <= 3 ) =  1.0 

(%i24) "P(ew-3*sigma <= X <= ew+3*sigma) (3-Sigma-Umgebung)"$

(%i25) sumBin(bin_n,ceiling(bin_ew-3*bin_sigma), floor(bin_ew+3*bin_sigma), bin_p)$
P( 0 <= X <= 3 ) =  1.0 

(%i26) xy:makelist([x,binomial(bin_n, x)*bin_p^x*bin_q^(bin_n-x)],x,0,bin_n);
(%o26)         [[0, 0.125], [1, 0.375], [2, 0.375], [3, 0.125]]

(%i28) "d)"$

(%i29) "ges.: P(A) = P(X=0) + P(X=3)"$

(%i30) "P(A) = P(X=0) + P(X=3) = 0.125 + 0.125 = 0.25"$

(%i31) "e)"$

(%i32) "B(n;p;k) mit n=3, p=0,4 und 0<=k<=3"$

(%i33) "Parameter der Binomialverteilung"$

(%i34) bin_n : 3$

(%i35) bin_p : 0.4$

(%i36) bin_q : 1-bin_p;
(%o36)                                0.6

(%i37) "Erwartungswert ew"$

(%i38) bin_ew : bin_n * bin_p;
(%o38)                                1.2

(%i39) "Standardabweichung sigma"$

(%i40) bin_sigma : ev(sqrt(bin_n * bin_p * bin_q),numer);
(%o40)                         0.84852813742386


(%i42) "P(ew-sigma <= X <= ew+sigma) (1-Sigma-Umgebung)"$

(%i43) sumBin(bin_n,ceiling(bin_ew-bin_sigma), floor(bin_ew+bin_sigma), bin_p)$
P( 1 <= X <= 2 ) =  0.72 

(%i44) "P(ew-2*sigma <= X <= ew+2*sigma) (2-Sigma-Umgebung)"$

(%i45) sumBin(bin_n,ceiling(bin_ew-2*bin_sigma), floor(bin_ew+2*bin_sigma), bin_p)$
P( 0 <= X <= 2 ) =  0.936 

(%i46) "P(ew-3*sigma <= X <= ew+3*sigma) (3-Sigma-Umgebung)"$

(%i47) sumBin(bin_n,ceiling(bin_ew-3*bin_sigma), floor(bin_ew+3*bin_sigma), bin_p)$
P( 0 <= X <= 3 ) =  1.0 

(%i48) xy:makelist([x,binomial(bin_n, x)*bin_p^x*bin_q^(bin_n-x)],x,0,bin_n);
(%o48)         [[0, 0.216], [1, 0.432], [2, 0.288], [3, 0.064]]

(%i50) "P(A) = P(X=0) + P(X=3) = 0.216 + 0.064"$

(%i51) 0.216 + 0.064;
(%o51)                               0.28

(%i52) "P(A) = P(X=0) + P(X=3) = 0.216 + 0.064 = 0.28"$