A ticket for an amusement park is going to be given, in a prize draw, to one of the $$80$$ members of a club. Of them, $$12$$ are blond, $$17$$ wear glasses, and $$4$$ are blond and use glasses.
1) Calculate the probability that the prize goes to someone who is not blond and does not use glasses.
2) If the one that receives the ticket is blond: what is the probability that he or she does not use glasses?
Development:
1) First, we create a contingency table to represent the information. We have two events, $$R =$$ "to be a blond" and $$G =$$ "to use glasses".
We first of all put in all the information of the statement.
Glasses | Without glasses | Total $$G,\overline{G}$$ | |
Blond | 4 | 12 | |
No Blond | |||
Total $$R,\overline{R}$$ | 17 | 80 |
Now we complete it. We know that there are in total $$12$$ blonds, and only $$4$$ use glasses. Therefore, the number of blonds who do not use glasses is $$12-4=8$$. Also, there are in total $$17$$ that use glasses, and only $$4$$ are blond and use glasses, therefore $$17-4 = 13$$ are not blond and use glasses.
On the other hand , in the club there is are a total of $$80$$ people. Therefore, if $$17$$ use glasses, $$80-17=63$$ do not use them. If $$12$$ pepole are blond, then $$80-12=68$$ people are not.
We introduce all this information in the contingency table.
Glasses | Without glasses | Total $$G,\overline{G}$$ | |
Blond | 4 | 8 | 12 |
No Blond | 13 | 68 | |
Total $$R,\overline{R}$$ | 17 | 63 | 80 |
Finally, if there are $$63$$ people in total who do not use glasses, and $$8$$ of them are blond, then $$63-8=55$$ are blond and do not use glasses.
Also we might have calculated it from another point of view: if there are $$68$$ people who are not blond, and $$13$$ of them use glasses, then $$68-13=55$$ people are not blond and do not use glasses.
And so, we already have our contingency table completed.
Glasses | Without glasses | Total $$G,\overline{G}$$ | |
Blond | 4 | 8 | 12 |
No Blond | 13 | 55 | 68 |
Total $$R,\overline{R}$$ | 17 | 63 | 80 |
With the table, we can already answer the question.
The probability we have been asked is $$P(\overline{R}\cap\overline{G})=\dfrac{55}{80}$$, by Laplace's law, since there are $$55$$ people with these characteristics out of the total $$80$$ (and to all of them it is equally probable to win the ticket).
2) We will calculate it with the formula of the conditional probabilities that we learned in the previous level. We want to calculate $$P(R/\overline{G})=\dfrac{P(R\cap\overline{G})}{P(\overline{G})}$$. Looking at the information in the table, $$$P(R\cap\overline{G})=\dfrac{8}{80}$$$ and $$$P(\overline{G})=\dfrac{63}{80}$$$ therefore $$$P(R/\overline{G})=\dfrac{\dfrac{8}{80}}{\dfrac{63}{80}}=\dfrac{8}{63}$$$
If we notice, since we want to calculate the probability that he or she does not wear glasses, we look at the column "without glasses" (vertical). In this column, $$8$$ are blond out of the total $$63$$, therefore the probability is $$\dfrac{8}{63}$$, again.
Solution:
1) $$P(\overline{R}\cap\overline{G})=\dfrac{55}{80}$$
2) $$P(R/\overline{G})=\dfrac{8}{63}$$