When a book is chosen from a library at random:
[tex]
\begin{array}{l}
P(\text { paperback })=0.46 \\
P(\text { fiction })=0.23
\end{array}
[/tex]
[tex]P(\text{both paperback and fiction}) = 0.17[/tex]
Draw a Venn diagram to work out [tex]P(\text{neither paperback nor fiction})[/tex] as a decimal.
Answer (2)
0.48
Explanation
Understand the problem and provided data We are given the following probabilities:
P ( paperback ) = 0.46
P ( fiction ) = 0.23
P ( both paperback and fiction ) = 0.17
We want to find the probability that a book is neither paperback nor fiction.
Find the probability of paperback or fiction First, we need to find the probability that a book is either paperback or fiction or both. We can use the formula for the probability of the union of two events:
P ( A or B ) = P ( A ) + P ( B ) − P ( A and B )
Apply the formula In our case, A is the event that a book is paperback, and B is the event that a book is fiction. So we have:
P ( paperback or fiction ) = P ( paperback ) + P ( fiction ) − P ( both paperback and fiction )
Calculate the probability of paperback or fiction Plugging in the given values, we get:
P ( paperback or fiction ) = 0.46 + 0.23 − 0.17 = 0.52
Find the probability of neither paperback nor fiction Now, we want to find the probability that a book is neither paperback nor fiction. This is the complement of the event that a book is either paperback or fiction. So we have:
P ( neither paperback nor fiction ) = 1 − P ( paperback or fiction )
Calculate the probability of neither paperback nor fiction Plugging in the value we found for P ( paperback or fiction ) , we get:
P ( neither paperback nor fiction ) = 1 − 0.52 = 0.48
State the final answer Therefore, the probability that a book is neither paperback nor fiction is 0.48 .
Examples
This type of probability calculation is useful in many real-world scenarios. For example, a marketing analyst might use this to determine the probability that a customer likes either product A or product B, or neither. This can help them tailor their marketing strategies to target specific customer segments. Similarly, a public health official might use this to determine the probability that a person has either disease X or disease Y, or neither, to help them allocate resources for disease prevention and treatment. The formula P ( A or B ) = P ( A ) + P ( B ) − P ( A and B ) is a fundamental tool in probability theory and has wide-ranging applications.
The probability that a book chosen from the library is neither paperback nor fiction is 0.48. To calculate this, we first find the probability of being either paperback or fiction and then use the complement rule. The final answer is 0.48 .
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