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1

2,5 puntos
Probabilidad y estadística
Durante la semana cultural de un centro docente, el Departamento de Matemáticas organiza un juego de azar por rondas en el que los alumnos pueden obtener puntos canjeables por pequeños premios. En cada ronda se lanza primero una moneda y, a continuación, un dado de seis caras. El resultado de la moneda combinado con el valor obtenido por el dado determina la condición para ganar la ronda: • Si la moneda sale cara, el alumno gana la ronda si el número obtenido en el dado es mayor o igual que 55. • Si la moneda sale cruz, el alumno gana la ronda si el número obtenido en el dado es par. Cada ronda es independiente y sigue exactamente este mismo procedimiento. Se pide:
1.1
Calcular la probabilidad de que, en una ronda cualquiera, el alumno gane.0,5 pts
1.2
Sabiendo que en una ronda el alumno ha ganado, ¿cuál es la probabilidad de que en el lanzamiento de la moneda haya salido cara?1 pt
1.3
Si se utiliza una moneda trucada tal que al lanzarla siempre se obtiene cara, y se realizan exactamente 77 rondas del juego, calcular la probabilidad de que el alumno gane exactamente 55 veces.1 pt

Formulario

Mostrar
Tabla de la distribución binomial (Bin⁡(n,p)\operatorname{Bin}(n,p))F(x)=P(X≤x)=∑k=0x(nk)pkqn−kF(x)=P(X\leq x)=\sum_{k=0}^{x}\binom{n}{k}p^kq^{n-k}
nnkkpp
0,010,05
100,99000,9500
11,00001,0000
200,98010,9025
10,99990,9975
21,00001,0000
300,97030,8574
10,99970,9928
21,00000,9999
31,00001,0000
400,96060,8145
10,99940,9860
21,00000,9995
31,00001,0000
41,00001,0000
500,95100,7738
10,99900,9774
21,00000,9988
31,00001,0000
41,00001,0000
51,00001,0000
600,94150,7351
10,99850,9672
21,00000,9978
31,00000,9999
41,00001,0000
51,00001,0000
61,00001,0000
700,93210,6983
10,99800,9556
21,00000,9962
31,00000,9998
41,00001,0000
51,00001,0000
61,00001,0000
71,00001,0000
800,92270,6634
10,99730,9428
20,99990,9942
31,00000,9996
41,00001,0000
51,00001,0000
61,00001,0000
71,00001,0000
81,00001,0000
900,91350,6302
10,99660,9288
20,99990,9916
31,00000,9994
41,00001,0000
51,00001,0000
61,00001,0000
71,00001,0000
81,00001,0000
91,00001,0000
nnkkpp
0,100,20
100,90000,8000
11,00001,0000
200,81000,6400
10,99000,9600
21,00001,0000
300,72900,5120
10,97200,8960
20,99900,9920
31,00001,0000
400,65610,4096
10,94770,8192
20,99630,9728
30,99990,9984
41,00001,0000
500,59050,3277
10,91850,7373
20,99140,9421
30,99950,9933
41,00000,9997
51,00001,0000
600,53140,2621
10,88570,6554
20,98420,9011
30,99870,9830
40,99990,9984
51,00000,9999
61,00001,0000
700,47830,2097
10,85030,5767
20,97430,8520
30,99730,9667
40,99980,9953
51,00000,9996
61,00001,0000
71,00001,0000
800,43050,1678
10,81310,5033
20,96190,7969
30,99500,9437
40,99960,9896
51,00000,9988
61,00000,9999
71,00001,0000
81,00001,0000
900,38740,1342
10,77480,4362
20,94700,7382
30,99170,9144
40,99910,9804
50,99990,9969
61,00000,9997
71,00001,0000
81,00001,0000
91,00001,0000
nnkkpp
0,250,30
100,75000,7000
11,00001,0000
200,56250,4900
10,93750,9100
21,00001,0000
300,42190,3430
10,84380,7840
20,98440,9730
31,00001,0000
400,31640,2401
10,73830,6517
20,94920,9163
30,99610,9919
41,00001,0000
500,23730,1681
10,63280,5282
20,89650,8369
30,98440,9692
40,99900,9976
51,00001,0000
600,17800,1176
10,53390,4202
20,83060,7443
30,96240,9295
40,99540,9891
50,99980,9993
61,00001,0000
700,13350,0824
10,44490,3294
20,75640,6471
30,92940,8740
40,98710,9712
50,99870,9962
60,99990,9998
71,00001,0000
800,10010,0576
10,36710,2553
20,67850,5518
30,88620,8059
40,97270,9420
50,99580,9887
60,99960,9987
71,00000,9999
81,00001,0000
900,07510,0404
10,30030,1960
20,60070,4628
30,83430,7297
40,95110,9012
50,99000,9747
60,99870,9957
70,99990,9996
81,00001,0000
91,00001,0000
nnkkpp
1/31/30,35
100,66670,6500
11,00001,0000
200,44440,4225
10,88890,8775
21,00001,0000
300,29630,2746
10,74070,7183
20,96300,9571
31,00001,0000
400,19750,1785
10,59260,5630
20,88890,8735
30,98770,9850
41,00001,0000
500,13170,1160
10,46090,4284
20,79010,7648
30,95470,9460
40,99590,9947
51,00001,0000
600,08780,0754
10,35120,3191
20,68040,6471
30,89990,8826
40,98220,9777
50,99860,9982
61,00001,0000
700,05850,0490
10,26340,2338
20,57060,5323
30,82670,8002
40,95470,9444
50,99310,9910
60,99950,9994
71,00001,0000
800,03900,0319
10,19510,1691
20,46820,4278
30,74140,7064
40,91210,8939
50,98030,9747
60,99740,9964
70,99980,9998
81,00001,0000
900,02600,0207
10,14310,1211
20,37720,3373
30,65030,6089
40,85520,8283
50,95760,9464
60,99170,9888
70,99900,9986
80,99990,9999
91,00001,0000
nnkkpp
0,400,45
100,60000,5500
11,00001,0000
200,36000,3025
10,84000,7975
21,00001,0000
300,21600,1664
10,64800,5748
20,93600,9089
31,00001,0000
400,12960,0915
10,47520,3910
20,82080,7585
30,97440,9590
41,00001,0000
500,07780,0503
10,33700,2562
20,68260,5931
30,91300,8688
40,98980,9815
51,00001,0000
600,04670,0277
10,23330,1636
20,54430,4415
30,82080,7447
40,95900,9308
50,99590,9917
61,00001,0000
700,02800,0152
10,15860,1024
20,41990,3164
30,71020,6083
40,90370,8471
50,98120,9643
60,99840,9963
71,00001,0000
800,01680,0084
10,10640,0632
20,31540,2201
30,59410,4770
40,82630,7396
50,95020,9115
60,99150,9819
70,99930,9983
81,00001,0000
900,01010,0046
10,07050,0385
20,23180,1495
30,48260,3614
40,73340,6214
50,90060,8342
60,97500,9502
70,99620,9909
80,99970,9992
91,00001,0000
nnkkpp
0,50
100,5000
11,0000
200,2500
10,7500
21,0000
300,1250
10,5000
20,8750
31,0000
400,0625
10,3125
20,6875
30,9375
41,0000
500,0313
10,1875
20,5000
30,8125
40,9688
51,0000
600,0156
10,1094
20,3438
30,6563
40,8906
50,9844
61,0000
700,0078
10,0625
20,2266
30,5000
40,7734
50,9375
60,9922
71,0000
800,0039
10,0352
20,1445
30,3633
40,6367
50,8555
60,9648
70,9961
81,0000
900,0020
10,0195
20,0898
30,2539
40,5000
50,7461
60,9102
70,9805
80,9980
91,0000
nnkkpp
0,010,05
1000,90440,5987
10,99570,9139
20,99990,9885
31,00000,9990
41,00000,9999
51,00001,0000
61,00001,0000
71,00001,0000
81,00001,0000
91,00001,0000
101,00001,0000
1100,89530,5688
10,99480,8981
20,99980,9848
31,00000,9984
41,00000,9999
51,00001,0000
61,00001,0000
71,00001,0000
81,00001,0000
91,00001,0000
101,00001,0000
111,00001,0000
1200,88640,5404
10,99380,8816
20,99980,9804
31,00000,9978
41,00000,9998
51,00001,0000
61,00001,0000
71,00001,0000
81,00001,0000
91,00001,0000
101,00001,0000
111,00001,0000
121,00001,0000
1300,87750,5133
10,99280,8646
20,99970,9755
31,00000,9969
41,00000,9997
51,00001,0000
61,00001,0000
71,00001,0000
81,00001,0000
91,00001,0000
101,00001,0000
111,00001,0000
121,00001,0000
131,00001,0000
nnkkpp
0,100,20
1000,34870,1074
10,73610,3758
20,92980,6778
30,98720,8791
40,99840,9672
50,99990,9936
61,00000,9991
71,00000,9999
81,00001,0000
91,00001,0000
101,00001,0000
1100,31380,0859
10,69740,3221
20,91040,6174
30,98150,8389
40,99720,9496
50,99970,9883
61,00000,9980
71,00000,9998
81,00001,0000
91,00001,0000
101,00001,0000
111,00001,0000
1200,28240,0687
10,65900,2749
20,88910,5583
30,97440,7946
40,99570,9274
50,99950,9806
60,99990,9961
71,00000,9994
81,00000,9999
91,00001,0000
101,00001,0000
111,00001,0000
121,00001,0000
1300,25420,0550
10,62130,2336
20,86610,5017
30,96580,7473
40,99350,9009
50,99910,9700
60,99990,9930
71,00000,9988
81,00000,9998
91,00001,0000
101,00001,0000
111,00001,0000
121,00001,0000
131,00001,0000
nnkkpp
0,250,30
1000,05630,0282
10,24400,1493
20,52560,3828
30,77590,6496
40,92190,8497
50,98030,9527
60,99650,9894
70,99960,9984
81,00000,9999
91,00001,0000
101,00001,0000
1100,04220,0198
10,19710,1130
20,45520,3127
30,71330,5696
40,88540,7897
50,96570,9218
60,99240,9784
70,99880,9957
80,99990,9994
91,00001,0000
101,00001,0000
111,00001,0000
1200,03170,0138
10,15840,0850
20,39070,2528
30,64880,4925
40,84240,7237
50,94560,8822
60,98570,9614
70,99720,9905
80,99960,9983
91,00000,9998
101,00001,0000
111,00001,0000
121,00001,0000
1300,02380,0097
10,12670,0637
20,33260,2025
30,58430,4206
40,79400,6543
50,91980,8346
60,97570,9376
70,99440,9818
80,99900,9960
90,99990,9993
101,00000,9999
111,00001,0000
121,00001,0000
131,00001,0000
nnkkpp
1/31/30,35
1000,01730,0135
10,10400,0860
20,29910,2616
30,55930,5138
40,78690,7515
50,92340,9051
60,98030,9740
70,99660,9952
80,99960,9995
91,00001,0000
101,00001,0000
1100,01160,0088
10,07510,0606
20,23410,2001
30,47260,4256
40,71100,6683
50,87790,8513
60,96140,9499
70,99120,9878
80,99860,9980
90,99990,9998
101,00001,0000
111,00001,0000
1200,00770,0057
10,05400,0424
20,18110,1513
30,39310,3467
40,63150,5833
50,82230,7873
60,93360,9154
70,98120,9745
80,99610,9944
90,99950,9992
101,00000,9999
111,00001,0000
121,00001,0000
1300,00510,0037
10,03850,0296
20,13870,1132
30,32240,2783
40,55200,5005
50,75870,7159
60,89650,8705
70,96530,9538
80,99120,9874
90,99840,9975
100,99980,9997
111,00001,0000
121,00001,0000
131,00001,0000
nnkkpp
0,400,45
1000,00600,0025
10,04640,0233
20,16730,0996
30,38230,2660
40,63310,5044
50,83380,7384
60,94520,8980
70,98770,9726
80,99830,9955
90,99990,9997
101,00001,0000
1100,00360,0014
10,03020,0139
20,11890,0652
30,29630,1911
40,53280,3971
50,75350,6331
60,90060,8262
70,97070,9390
80,99410,9852
90,99930,9978
101,00000,9998
111,00001,0000
1200,00220,0008
10,01960,0083
20,08340,0421
30,22530,1345
40,43820,3044
50,66520,5269
60,84180,7393
70,94270,8883
80,98470,9644
90,99720,9921
100,99970,9989
111,00000,9999
121,00001,0000
1300,00130,0004
10,01260,0049
20,05790,0269
30,16860,0929
40,35300,2279
50,57440,4268
60,77120,6437
70,90230,8212
80,96790,9302
90,99220,9797
100,99870,9959
110,99990,9995
121,00001,0000
131,00001,0000
nnkkpp
0,50
1000,0010
10,0107
20,0547
30,1719
40,3770
50,6230
60,8281
70,9453
80,9893
90,9990
101,0000
1100,0005
10,0059
20,0327
30,1133
40,2744
50,5000
60,7256
70,8867
80,9673
90,9941
100,9995
111,0000
1200,0002
10,0032
20,0193
30,0730
40,1938
50,3872
60,6128
70,8062
80,9270
90,9807
100,9968
110,9998
121,0000
1300,0001
10,0017
20,0112
30,0461
40,1334
50,2905
60,5000
70,7095
80,8666
90,9539
100,9888
110,9983
120,9999
131,0000