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Probability — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSProbability4 Marks
Question
A class consists of 10 boys and 8 girls. Three students are selected at random. What is the probability that the selected group has
  1. All boys?
  2. All girls?
  3. 1 boys and 2 girls?
  4. At least one girl?
  5. At most one girl?
✓

Answer

10 boys 8 girls Three student are selected at random $\text{n}(\text{S})=^{18}\text{C}_3$
  1. E be the event that the group has all boys
$\therefore\text{n}(\text{E})=^{18}\text{C}_3$
$\therefore\text{p}(\text{E})=\frac{^{10}\text{C}_3}{^{18}\text{C}_3}$
$=\frac{10\times9\times8}{18\times17\times16}$
$=\frac{5}{34}$
  1. E be the event that the group has all girls
$\therefore\text{n}(\text{E})=^8\text{C}_3$
$\therefore\text{p}(\text{E})=\frac{^{8}\text{C}_3}{^{18}\text{C}_3}$
$=\frac{8\times7\times6}{18\times17\times16}$
$=\frac{7}{102}$
  1. E be the event that the group has one boy and two girls
$\therefore\text{n}(\text{E})=^8\text{C}_3\times^{10}\text{C}_2$
$\therefore\text{p}(\text{E})=\frac{^{8}\text{C}_1\times^{10}\text{C}_2}{^{18}\text{C}_3}$
$=\frac{35}{102}$
  1. E be the event that atleast one girls in the group
$\text{E}=(1,\ 2,\ 3)\text{girls}$
$\therefore\text{n}(\text{E})=^8\text{C}_1\times^{10}\text{C}_2+^8\text{C}_2\times^{10}\text{C}_1+^8\text{C}_3\times^{10}\text{C}_0$
$\text{p}(\text{E})=\frac{^{8}\text{C}_1\times^{10}\text{C}_2+^{8}\text{C}_2\times^{10}\text{C}_1+^{8}\text{C}_3}{^{18}\text{C}_3}$
$=\frac{29}{34}$
  1. E be the event that almost one girls in the group
$\text{E}=(0,\ 1,)\text{girls}$
$\therefore\text{n}(\text{E})=^8\text{C}_0\times^{10}\text{C}_3+^8\text{C}_1\times^{10}\text{C}_2$
$\text{p}(\text{E})=\frac{^{10}\text{C}_3+8\times^{10}\text{C}_2}{^{18}\text{C}_3}$
$=\frac{10}{17}$

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