Introduction to probability：练习题

A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. One marble is chosen at random.
What is the probability that the marble is not red?

The probability that it will rain on a particular day is \(0.65\).
What is the probability that it will not rain on that day?

A biased spinner has four sections coloured Red, Blue, Green, and Yellow. The probability of landing on Red is \(0.2\). The probability of landing on Blue is twice the probability of landing on Green. The probability of landing on Yellow is the same as the probability of landing on Red.
What is the probability of landing on Blue?

An ordinary fair six-sided die is rolled.
What is the probability of rolling a number greater than 4?

A spinner is spun 200 times. The results are shown in the following table:

\(\begin{array}{|l|c|c|c|c|} \hline \text{Color} & \text{Red} & \text{Blue} & \text{Yellow} & \text{Green} \\ \hline \text{Frequency} & 45 & 55 & 60 & 40 \\ \hline \end{array}\)

What is the relative frequency of the spinner landing on Blue?

A biased dice is rolled. The probability of rolling a \(6\) is \(0.2\). If the dice is rolled \(150\) times, calculate the expected frequency of not rolling a \(6\).

A fair six-sided die is rolled twice. What is the probability of rolling an even number on the first roll and a number greater than 4 on the second roll?

A bag contains 5 red balls, 3 blue balls, and 2 green balls. A ball is chosen at random from the bag. What is the probability that the ball is blue?

A box contains 12 beads: 4 red, 5 blue, and 3 yellow. A bead is chosen at random, its color is noted, and it is then returned to the box. A second bead is then chosen at random.

(a) Find the probability that both beads chosen are blue.

(b) Work out the probability that exactly one of the chosen beads is red.

(c) If this experiment of picking two beads is repeated 144 times, calculate the expected number of times that a red bead is followed by a yellow bead.

In a group of 30 students, 18 study Physics (\(P\)), 15 study Chemistry (\(C\)), and 4 study neither subject. This information is represented in a Venn diagram.

(a) Find the number of students in the following regions:
(i) \(n(P \cap C)\)
(ii) \(n(P \cap C')\)

(b) A student is chosen at random from the group. Find the probability that this student studies at most one of these subjects.

(c) Two students are chosen at random from the group, one after the other, without replacement. Find the probability that:
(i) Both students study Chemistry.
(ii) One student studies Physics only and the other student studies Chemistry only.