use for questions 21 - 24: out of 50 students, 24 are taking art and 30 are taking music. sixteen of the…

use for questions 21 - 24: out of 50 students, 24 are taking art and 30 are taking music. sixteen of the students are taking both art and music. if two different students from this group are chosen at random, find each probability. 21. both students take art. 22. both students take both art and music. 23. both students take music, but not art. 24. neither student takes music.

use for questions 21 - 24: out of 50 students, 24 are taking art and 30 are taking music. sixteen of the students are taking both art and music. if two different students from this group are chosen at random, find each probability. 21. both students take art. 22. both students take both art and music. 23. both students take music, but not art. 24. neither student takes music.

Answer

Explanation:

Step1: Calculate number of students for each case

  • Number of students taking only art: $24 - 16=8$.
  • Number of students taking only music: $30 - 16 = 14$.
  • Number of students taking neither art nor music: $50-(8 + 16+14)=12$.

Step2: Calculate probabilities using combination formula $P=\frac{C(n,k)}{C(N,k)}$, where $C(n,k)=\frac{n!}{k!(n - k)!}$, $n$ is the number of favorable students, $N = 50$ is the total number of students, and $k = 2$ is the number of students chosen.

Question 21

  • $n = 24$, $N=50$, $k = 2$. $C(24,2)=\frac{24!}{2!(24 - 2)!}=\frac{24\times23}{2\times1}=276$. $C(50,2)=\frac{50!}{2!(50 - 2)!}=\frac{50\times49}{2\times1}=1225$. $P=\frac{C(24,2)}{C(50,2)}=\frac{276}{1225}$.

Question 22

  • $n = 16$, $N = 50$, $k=2$. $C(16,2)=\frac{16!}{2!(16 - 2)!}=\frac{16\times15}{2\times1}=120$. $P=\frac{C(16,2)}{C(50,2)}=\frac{120}{1225}=\frac{24}{245}$.

Question 23

  • $n = 14$, $N = 50$, $k = 2$. $C(14,2)=\frac{14!}{2!(14 - 2)!}=\frac{14\times13}{2\times1}=91$. $P=\frac{C(14,2)}{C(50,2)}=\frac{91}{1225}=\frac{13}{175}$.

Question 24

  • $n=50 - 30=20$, $N = 50$, $k = 2$. $C(20,2)=\frac{20!}{2!(20 - 2)!}=\frac{20\times19}{2\times1}=190$. $P=\frac{C(20,2)}{C(50,2)}=\frac{190}{1225}=\frac{38}{245}$.

Answer:

  1. $\frac{276}{1225}$
  2. $\frac{24}{245}$
  3. $\frac{13}{175}$
  4. $\frac{38}{245}$