Respuesta :
She selected 6 Mystery books and 10 Non-fiction books, a total of 16 books.
You have to determine the probability that if she chooses 3 books at random, the three will be Non-fiction.
To calculate this you have to consider this scenario as choosing "without replacement" since it will make no sense for her to choose the same book 3 times.
So, the probability of choosing a Non-fiction book (N), can be expressed as:
[tex]P(N_1\cap N_2\cap N_3)=P(N_1)\cdot P(N_2)\cdot P(N_3)[/tex]The subindices 1, 2, 3 indicate the order that the books were chosen.
-The probability of the first book being Non-fiction can be calculated as the quotient between the number of Non-fiction books and the total number of books:
[tex]\begin{gathered} P(N_1)=\frac{nº\text{non}-\text{fiction}}{\text{total books}} \\ P(N_1)=\frac{10}{16} \end{gathered}[/tex]Once this book was choose, there are 15 books left to choose from, 6 of them are Mystery books and the remaining 9 are Non-fiction.
To determine the probability of the second book being Non-fiction, you have to use the information of the remaining books:
[tex]\begin{gathered} P(N_2)=\frac{nºremaining\text{ non-fiction}}{remaining\text{ books}} \\ P(N_2)=\frac{9}{15} \end{gathered}[/tex]Once this book was chosen, the remaining number of books is 14 and the remaining number of Non-fiction books is 8.
Using these values you can calculate the probability of the first book being Non-fiction
[tex]\begin{gathered} P(N_3)=\frac{nº\text{remaining non-fiction}}{remaining\text{ }books} \\ P(N_3)=\frac{8}{14} \end{gathered}[/tex]Next, you can calculate the probability that the 3 books chosen are Non-fictional:
[tex]P(N_1\cap N_2\cap N_3)=P(N_1)\cdot P(N_2)\cdot P(N_3)=\frac{10}{16}\cdot\frac{9}{15}\cdot\frac{8}{14}=\frac{3}{14}\approx0.2143[/tex]The probability of choosing 3 books at random that is Non-fiction is 3/147 or 0.2143.
