Respuesta :
We have to find the probability that the commitee of 4 is being formed by the 4 teachers.
We have 32 teachers from a total of 4+32+5 = 41 employees.
We can calculate the probability in steps:
1) We find the probability that a teacher is chosen for the first place in the commitee. In this case, the order does not matter, but we use the places as a way to express the steps.
We can calculate this probability as 32/41, as we have 32 teachers out of 41 employees.
2) Now, we calculate the probability that a teacher is chosen for the second place. As one teacher has been already selected, we are left with 31 teachers out of 40 employees.
Then, the probability is 31/40.
3) In the same way, for the third place, we have the probability that a teacher is chosen as 30/39.
4) For the fourth place the probability is 29/38.
We can then write the probability that all four places are occupied by teachers by multiplying this 4 probabilities:
[tex]\begin{gathered} P(\text{TTTT})=P(x_1=T)\cdot P(x_2=T)\cdot P(x_3=T)\cdot P(x_4=T) \\ P(\text{TTTT})=\frac{32}{41}\cdot\frac{31}{40}\cdot\frac{30}{39}\cdot\frac{29}{38} \\ P(\text{TTTT})=\frac{863040}{2430480} \end{gathered}[/tex]We can easily simplify this fraction using the fractions we already have and grouping them as:
[tex]\begin{gathered} P=\frac{32}{41}\cdot\frac{31}{40}\cdot\frac{30}{39}\cdot\frac{29}{38} \\ P=\frac{2^5}{41}\cdot\frac{31}{2^3\cdot5}\cdot\frac{3\cdot2\cdot5}{3\cdot13}\cdot\frac{29}{2\cdot19} \\ P=\frac{2^6\cdot3\cdot5\cdot29\cdot31}{2^4\cdot3\cdot5\cdot13\cdot19\cdot41} \\ P=\frac{2^2\cdot29\cdot31}{13\cdot19\cdot41} \\ P=\frac{3596}{10127} \end{gathered}[/tex]Answer: the probability is 3596/10127 or 0.355090.
