Probability and conditional probability：練習問題

A fair six-sided die is rolled once. What is the probability that the result is either an even number or a number greater than 4?

A bag contains 4 white balls and 6 black balls. Two balls are drawn one after the other without replacement. What is the probability that both balls drawn are white?

In a certain population, 20% of the individuals possess a specific genetic trait. If two individuals are selected from this population at random, what is the probability that at least one of them possesses the genetic trait?

A box contains 20 chocolates: 8 are dark chocolates and 12 are milk chocolates. If one chocolate is selected at random, what is the probability that it is a dark chocolate?

A local theater offers three types of seats: Front Row, Middle Row, and Back Row. The distribution of tickets sold for a Friday night show is given in the table below:

\( \begin{array}{|l|c|c|c|} \hline & \text{Sold to Adults} & \text{Sold to Children} & \text{Total} \\ \hline \text{Front Row} & 45 & 15 & 60 \\ \hline \text{Middle Row} & 80 & 40 & 120 \\ \hline \text{Back Row} & 15 & 5 & 20 \\ \hline \text{Total} & 140 & 60 & 200 \\ \hline \end{array} \)

If a ticket is selected at random from those sold to adults, what is the probability that it is a Middle Row ticket?

A bag contains \( 12 \) red marbles and \( 8 \) blue marbles. If two marbles are selected at random one after the other without replacement, what is the probability, expressed as a fraction, that the first marble is red and the second marble is blue?

A large clinic tested \( 500 \) patients for a specific medical condition. The results are summarized in the following table structure:
- Of the \( 100 \) patients who actually have the condition, \( 92 \) tested positive.
- Of the \( 400 \) patients who do not have the condition, \( 20 \) tested positive.

Part A: If a patient is selected at random from the \( 500 \) patients, what is the probability that the patient tested positive given that they do not have the condition?
Part B: If a patient is selected at random from those who tested positive, what is the probability, rounded to the nearest hundredth, that the patient actually has the condition?
Part C: Explain how the probability in Part B would change if the number of false positives (those who do not have the condition but tested positive) increased.