There are 50 books on a shelf. They include 15 mystery novels, 12 works of historical fiction, 8 romance novels, and 15 nonfiction books. If Federica randomly selects 2 books from the shelf, what is the probability of selecting a historical fiction book on the first selection and selecting romance on the second selection? Assume the books are taken without replacement.(1 point)

Responses

850
Start Fraction 8 over 50 End Fraction

1250
Start Fraction 12 over 50 End Fraction

1250⋅850
Start Fraction 12 over 50 End Fraction times Start Fraction 8 over 50 End Fraction

1250⋅849
Start Fraction 12 over 50 End Fraction times Start Fraction 8 over 49 End Fraction

1 answer

To find the probability of selecting a historical fiction book first and a romance novel second without replacement, we can follow these steps:

  1. Calculate the probability of selecting a historical fiction book first.
    There are 12 historical fiction books out of a total of 50 books. Therefore, the probability of selecting a historical fiction book first is:

    \[ P(\text{Historical Fiction first}) = \frac{12}{50} \]

  2. Calculate the probability of selecting a romance novel second after having selected a historical fiction book first.
    After selecting the historical fiction book, there are now 49 books left on the shelf, and still 8 romance novels. Therefore, the probability of selecting a romance novel second is:

    \[ P(\text{Romance second | Historical Fiction first}) = \frac{8}{49} \]

  3. Calculate the combined probability of both events occurring.
    The total probability of both selecting a historical fiction book first and then a romance novel second is the product of the individual probabilities:

    \[ P(\text{Historical Fiction first and Romance second}) = P(\text{Historical Fiction first}) \times P(\text{Romance second | Historical Fiction first}) \]

    \[ = \frac{12}{50} \times \frac{8}{49} \]

    \[ = \frac{12 \times 8}{50 \times 49} \]

    \[ = \frac{96}{2450} \]

Looking at the options provided, the closest match to our calculations can be represented as:

\[ \frac{12}{50} \times \frac{8}{49} \]

This corresponds to the option:

Start Fraction 12 over 50 End Fraction times Start Fraction 8 over 49 End Fraction.