Exercise 6A: Combining Events

  1. A bag contains 8 blue and 6 red balls. If one ball is picked at random, find the probability that it is:

    1. A blue ball.

    2. A red ball.

  2. If \(P(A) = 0.4\), \(P(B) = 0.5\), and \(A\) and \(B\) are independent events, find \(P(A \cap B)\).

  3. In a group of 40 students, 25 like Mathematics, 20 like Science, and 10 like both. Find the probability that a student chosen at random:

    1. Likes Mathematics or Science.

    2. Likes neither subject.

Exercise 6B: Conditional Events

  1. Given \(P(A) = 0.7\), \(P(B) = 0.4\), and \(P(A \cap B) = 0.28\), are \(A\) and \(B\) independent?

  2. If \(P(A) = 0.6\) and \(P(B|A) = 0.5\), find \(P(A \cap B)\).

  3. A box contains 5 white and 3 black marbles. Two marbles are drawn one after the other without replacement. Find the probability that:

    1. Both are white.

    2. The second is black, given that the first was white.

Exercise 6C: Tree Diagrams

  1. A coin is tossed twice. Draw a tree diagram and find the probability of getting:

    1. Two heads.

    2. At least one tail.

  2. The probability that it rains on any day is 0.3. If it rains, the probability that Ahmed goes to the market is 0.4. If it does not rain, the probability is 0.8. Find the probability that Ahmed goes to the market.

  3. A bag contains 3 red and 7 blue pens. Two pens are selected at random without replacement. Use a tree diagram to find the probability that they are of different colors.

Exercise 6D: Counting Principles and Permutations

  1. In how many ways can 4 people be seated in a row of 4 chairs?

  2. How many different 3-letter codes can be formed using the letters \(\{A, B, C, D, E\}\) if repetition is not allowed?

  3. Find the value of:

    1. \(6!\)

    2. \(^{10}P_{2}\)

  4. How many ways can the letters of the word “MOGADISHU” be arranged?

Exercise 6E: Combinations

  1. A committee of 4 members is to be selected from 9 people. In how many ways can this be done?

  2. Find the value of:

    1. \(^{7}C_{3}\)

    2. \(^{5}C_{5}\)

  3. A box contains 6 red and 4 green balls. In how many ways can 3 balls be selected such that 2 are red and 1 is green?

  4. In a test, a student must choose 5 questions out of 8. How many choices does the student have?