條件概率（高階）：練習題

A bag contains only red, blue, and yellow counters.
The ratio of red counters to blue counters is \(3 : 5\).
The probability of picking a yellow counter is \(0.2\).
If there are 24 blue counters in the bag, calculate the total number of counters in the bag.

A bag contains \(n\) counters, of which 5 are red and the rest are blue.
Two counters are taken at random from the bag without replacement.
The probability that both counters are red is \(\frac{5}{33}\).
Find the total number of counters, \(n\), in the bag.

A bag contains only red, blue, and green marbles.
The probability of picking a red marble is \(0.3\).
The probability of picking a blue marble is \(0.45\).
What is the probability of picking a green marble?

A bag contains 5 red balls and \(y\) blue balls.
Two balls are taken at random without replacement.
The probability that one ball is red and the other is blue is \(\frac{1}{2}\).
Find the possible values of \(y\).

In a group of 100 people, 70 like tea (T), 60 like coffee (C), and 25 like neither.
A person is chosen at random from the group.
Find the probability that this person likes both tea and coffee.

A bag contains only red, blue, and green counters.
The probability of picking a red counter is 0.3.
The probability of picking a blue counter is 0.4.
Find the probability of picking a green counter.

In a class of 30 students, 18 study Art (\( A \)) and 15 study Biology (\( B \)). 7 students study both subjects.
Given that a student selected at random studies Art, find the probability that they also study Biology.

A bag contains x red counters and (x + 3) blue counters. Two counters are taken at random from the bag without replacement.
The probability that both counters are the same colour is \(\frac{1}{2}\).
Form an equation in terms of \(x\) and solve it to find the total number of counters in the bag.

A bag contains \(6\) red counters and \(4\) blue counters. Jane takes at random a counter from the bag and notes its colour. She does not replace the counter. She then takes at random a second counter from the bag.

(a) Calculate the probability that Jane takes two counters of the same colour.
(b) Calculate the probability that Jane takes at least one blue counter.

A bag contains \( n \) counters. 7 of the counters are green and the rest are blue.
Two counters are taken at random from the bag without replacement.
The probability that both counters are green is \( \frac{7}{15} \).

(a) Show that \( n^2 - n - 90 = 0 \).
(b) Find the value of \( n \) and hence find the number of blue counters in the bag.