Answer to Question #98186 in Statistics and Probability for Emmanuel Imion

Question

probability question: a class survey carried out in a school showed that 63â„… of the students played sports, 27â„… played musical instruments while 20â„… played neither sports musical instruments. is the event exclusive? Find the probability of a student playing sport or musical instruments or both. Find the probability of playing both musical instrument and sport

Answer 1

Let $S=$ the event "student is playing sport", $M=$ the event "student is playing musical instruments"

Two events $A$ and $B$ are mutually exclusive if they cannot occur at the same time

$$P(A\cap B)=0$$

Given that $P(S)=0.63, P(M)=0.27, P(\overline{S\cup M})=0.2$

Find the probability of a student playing sport or musical instruments or both

$$P(S\cup M)=1-P(\overline{S\cup M})=1-0.2=0.8$$

The probability of a student playing sport or musical instruments or both is $0.8$

Find the probability of playing both musical instrument and sport

$$P(S\cup M)=P(S)+P(M)-P(S\cap M)$$

$$P(S\cap M)=P(S)+P(M)-P(S\cup M)$$

$$P(S\cap M)=0.63+0.27-0.8=0.1$$

The probability of a student playing both musical instrument and sport is $0.1$

Hence the events "student is playing sport" and "student is playing musical instruments" are not mutually exclusive.