Toevalsvariabelen

Antwoorden bij de opgaven

1. 1. ${\displaystyle x}$	0	1	2	3	4
      ${\displaystyle P(X=x)}$	0,3164	0,4219	0,2109	0,0469	0,0039

   2. 1

2. 1. ${\displaystyle x}$	0	1	2	3
      ${\displaystyle P(X=x)}$	${\displaystyle \frac{125}{216}}$	${\displaystyle \frac{75}{216}}$	${\displaystyle \frac{15}{216}}$	${\displaystyle \frac{1}{216}}$

   2. 0,5
   3. Als je heel vaak met drie dobbelstenen gooit, mag je per 2 keer gooien een zes verwachten.
3. 1. -
   2. ${\displaystyle \frac{304}{729}}$
   3. ${\displaystyle 1\frac{2}{3}}$
   4. De verwachting van de mannen was ${\displaystyle 1\frac{1}{3}}$ en er zijn 3 taken uit te voeren.

4. 1. aantal zaden	3	4	5	6	7	8	9	10	11
      frequentie	0,0056	0,0112	0,0449	0,0730	0,1236	0,2528	0,3539	0,1292	0,0056

   2. 0,4887 dus ongeveer 49%
   3. 0,0168
   4. 8.15

5. 1. ${\displaystyle j}$	0	1	2	3	4
      ${\displaystyle P(J=j)}$	${\displaystyle \frac{1}{16}}$	${\displaystyle \frac{4}{16}}$	${\displaystyle \frac{6}{16}}$	${\displaystyle \frac{4}{16}}$	${\displaystyle \frac{1}{16}}$

   2. Verwachtingswaarde is 2
   3. 300

6. 1. ${\displaystyle x}$	0	1	2	3	4
      ${\displaystyle \text{P}(X=x)}$	0,6830	0,2732	0,0410	0,0027	0,0001

   2. ${\displaystyle y}$	0	1	2	3
      ${\displaystyle \text{P}(Y=y)}$	0,6697	0,2977	0,0319	0,0007

7. 1. 0, 1, 2, 3, 4

   2. ${\displaystyle m}$	0	1	2	3	4
      ${\displaystyle \text{P}(M=m)}$	0,0242	0,1719	0,3868	0,3283	0,0889

   3. ongeveer 1.3

8. 1. ${\displaystyle z}$	1	2	3	4	5	6	8	9	10	12	15	16	18	20	24	25	30	36
      ${\displaystyle \text{P}(Z=z)}$	${\displaystyle \frac{1}{36}}$	${\displaystyle \frac{2}{36}}$	${\displaystyle \frac{2}{36}}$	${\displaystyle \frac{3}{36}}$	${\displaystyle \frac{2}{36}}$	${\displaystyle \frac{4}{36}}$	${\displaystyle \frac{2}{36}}$	${\displaystyle \frac{1}{36}}$	${\displaystyle \frac{2}{36}}$	${\displaystyle \frac{4}{36}}$	${\displaystyle \frac{2}{36}}$	${\displaystyle \frac{1}{36}}$	${\displaystyle \frac{2}{36}}$	${\displaystyle \frac{2}{36}}$	${\displaystyle \frac{2}{36}}$	${\displaystyle \frac{1}{36}}$	${\displaystyle \frac{2}{36}}$	${\displaystyle \frac{1}{36}}$

   2. De verwachtingswaarde is 12,25, dus 12,00 per spel inzetten levert gemiddeld winst op. De winstkans bij n spel is ${\displaystyle \frac{13}{36}}$.
9. 1. ${\displaystyle P(S=3)=\mathrm{0,25};P(S=4)=\mathrm{0,375}}$ en ${\displaystyle P(S=5)=\mathrm{0,375}}$. De verwachtingswaarde is 4,125 sets
   2. 412 of 413 sets
   3. ${\displaystyle P(S=3)\approx \mathrm{0,49};P(S=4)\approx \mathrm{0,24}}$ en ${\displaystyle P(S=5)\approx \mathrm{0,27}}$. De verwachtingswaarde is ongeveer 3,8 sets
   4. ${\displaystyle P(S=3)=\mathrm{0,49};P(S=4)=\mathrm{0,273}}$ en ${\displaystyle P(S=5)=\mathrm{0,237}}$
   5. Ongeveer 3,75

10. 1. ${\displaystyle y}$	0	1	2	4	8	16	32
       ${\displaystyle \text{P}(Y=y)}$	0,015625	0,5	0,25	0,125	0,0625	0,03125	0,015625

    2. 3 euro
    3. 0,046875
11. 1. ${\displaystyle P(M=0)\approx \mathrm{12,17}\%}$ en ${\displaystyle P(M=1)\approx \mathrm{37,46}\%}$
    2. ${\displaystyle P(M=2)\approx \mathrm{36,12}\%;P(M=3)\approx \mathrm{12,84}\%}$ en ${\displaystyle P(M=4)\approx \mathrm{1,40}\%}$
    3. ${\displaystyle P(V=0)=P(M=4)}$ enzovoorts.
    4. 1,54 en 2,46
12. 1. ${\displaystyle P(Z=0)=\frac{1}{35};P(Z=1)=\frac{12}{35};P(Z=2)=\frac{18}{35}}$ en ${\displaystyle P(Z=3)=\frac{4}{35}}$
    2. ${\displaystyle \frac{22}{35}}$
    3. ${\displaystyle 1\frac{5}{7}}$

13. 1. ${\displaystyle k}$	0	1	2	3	4
       ${\displaystyle \text{P}(K=k)}$	0,0625	0,25	0,375	0,75	0,0625

    2. 2
    3. 50 keer
14. 1. 6.4%
    2. P(cat.I)=0,064; P(cat.II)=0,288; P(cat.III)=0,432 en P(cat.IV)=0,216. Dus de meeste in categorie III
    3. 72