Answer to Question #280135 in Statistics and Probability for muqu

Question

In a sample of 100 students enrolling in a university, a questionnaire indicated that 45 of them studied Fashion, 40 studied Textile, 35 studied Drafting, 20 studied both Fashion and Textile, 23 studied both Fashion and Drafting, 19 studied both Textile and Drafting, and 12 studied all three languages.

 

Using a Venn diagram, find the probability that a randomly chosen student studied only one of the three languages.

 

Answer 1

$$N(F\cup T \cup D)=N(F)+N(T)+N(D)$$

$$-N(F\cap T)-N(F\cap D)-N(T\cap D)$$

$$+N(F\cap T\cap D)=45+40+35$$

$$-20-23-19+12=70$$

$$N((F\cup T \cup D)^C)=N(U)-N(F\cup T \cup D)$$

$$=100-70=30$$

$$N((F\cap T)-D)=N(F\cap T)-N(F\cap T\cap D)$$

$$=20-12=8$$

$$N((F\cap D)-T)=N(F\cap D)-N(F\cap T\cap D)$$

$$=23-12=11$$

$$N((T\cap D)-F)=N(T\cap D)-N(F\cap T\cap D)$$

$$=19-12=7$$

$$N(only\ F)=45-8-11-12=14$$

$$N(only\ T)=40-8-7-12=13$$

$$N(only\ D)=35-7-11-12=5$$

$$P(only\ one)=\dfrac{14+13+5}{100}=0.32$$