1. On a single toss of one die, find the probability of obtaining a) The number 4 b) an odd number c) an even number d) a number less than 4 e) a number greater than 4 f) an odd or an even number 2. In a junior school class of 28 pupils, 7 are in both a sports team and the school band. There are 16 pupils involved in sports teams and 10 in the school band. Find the probability that a pupil chosen at random. a) is only in the school band b) is in either a sports team or the school band c) is in neither a sports team nor the school band 3. An integer is selected randomly from a set of integers \( \{1,2,3,4,5,6,7,8,9,10,11,12\} \). Find the probability that the integer is a) an even number or is divisible by 3 b) an even number and is not divisible by 3 c) not an even number and is not divisibleby 3 . 4. For the sample space \( S \), it is given that \[ \begin{array}{c} P(A)=0.5 \\ P(A \cup B)=0.6 \\ P(A \cap B)=0.2 \end{array} \] Find: a) \( P(B) \) b) \( P\left(A^{\prime} \cap B\right) \) c) \( P\left(A \cap B^{\prime}\right) \) d) \( P\left(A^{\prime} \cap B^{\prime}\right) \) 5. \( \quad C \) and \( D \) are two events where \( P(C)=0.1, P(D)=0.2 \) and \( P(C \cup D)=0.3 \). a) Determine whether \( C \) and \( D \) are two mutually exclusive events. b) Find \( P\left(C^{\prime}\right) \) and \( P\left(C^{\prime} \cap D^{\prime}\right) \) 6. The events \( A, B, C \) and \( D \) are mutually exclusive with \( P(A)=P(B)=0.3 \) and \( P(C)= \) \( P(D)=0.1 \). If \( E \) and \( F \) are events defined by \( E=A \cup D \) and \( F=B \cup C \), find a) \( P(E \cup F) \) b) \( P(E \cap F) \)

Let's solve each part of the problem step by step. ### 1. Probability of outcomes from a single die toss A standard die has 6 faces numbered from 1 to 6. **a)** Probability of obtaining the number 4: - There is 1 favorable outcome (4). - Total outcomes = 6. \[ P(4) = \frac{1}{6} \] **b)** Probability of obtaining an odd number: - Odd numbers on a die: 1, 3, 5 (3 favorable outcomes). \[ P(\text{odd}) = \frac{3}{6} = \frac{1}{2} \] **c)** Probability of obtaining an even number: - Even numbers on a die: 2, 4, 6 (3 favorable outcomes). \[ P(\text{even}) = \frac{3}{6} = \frac{1}{2} \] **d)** Probability of obtaining a number less than 4: - Numbers less than 4: 1, 2, 3 (3 favorable outcomes). \[ P(<4) = \frac{3}{6} = \frac{1}{2} \] **e)** Probability of obtaining a number greater than 4: - Numbers greater than 4: 5, 6 (2 favorable outcomes). \[ P(>4) = \frac{2}{6} = \frac{1}{3} \] **f)** Probability of obtaining an odd or an even number: - Since all outcomes are either odd or even, the probability is: \[ P(\text{odd or even}) = 1 \] ### 2. Probability in a junior school class Let: - \( n(S) = 28 \) (total pupils) - \( n(T) = 16 \) (pupils in sports teams) - \( n(B) = 10 \) (pupils in the school band) - \( n(T \cap B) = 7 \) (pupils in both) **a)** Probability that a pupil is only in the school band: \[ n(B \text{ only}) = n(B) - n(T \cap B) = 10 - 7 = 3 \] \[ P(B \text{ only}) = \frac{3}{28} \] **b)** Probability that a pupil is in either a sports team or the school band: \[ n(T \cup B) = n(T) + n(B) - n(T \cap B) = 16 + 10 - 7 = 19 \] \[ P(T \cup B) = \frac{19}{28} \] **c)** Probability that a pupil is in neither a sports team nor the school band: \[ n(\text{neither}) = n(S) - n(T \cup B) = 28 - 19 = 9 \] \[ P(\text{neither}) = \frac{9}{28} \] ### 3. Probability from a set of integers The set is \( \{1,2,3,4,5,6,7,8,9,10,11,12\} \) (12 integers). **a)** Probability that the integer is an even number or is divisible by 3: - Even numbers: 2, 4, 6, 8, 10, 12 (6 outcomes). - Divisible by 3: 3, 6, 9, 12 (4 outcomes). - Common (even and divisible by 3): 6, 12 (2 outcomes). \[ P(\text{even or divisible by 3}) = \frac{6 + 4 - 2}{12} = \frac{8}{12} = \frac{2}{3} \] **b)** Probability that the integer is an even number and is not divisible by 3: - Even numbers not divisible by 3: 2, 4, 8, 10 (4 outcomes). \[ P(\text{even and not divisible by 3}) = \frac{4}{12} = \frac{1}{3} \] **c)** Probability that the integer is not an even number and is not divisible by 3: - Odd numbers: 1, 3, 5, 7, 9, 11 (6 outcomes). - Not divisible by 3: 1, 5, 7, 11 (4 outcomes). \[ P(\text{not even and not divisible by 3}) = \frac{4}{12} = \frac{1}{3} \] ### 4. Given probabilities Given: \[ P(A) = 0.5, \quad P(A \cup B) = 0.6, \quad P(A \cap B) = 0.2 \] **a)** Find \( P(B) \): \[ P(B) = P(A \cup B) - P(A) + P(A \cap B) = 0.6 - 0.5 + 0.2 = 0.3 \] **b)** Find \( P(A^{\prime} \cap B) \): \[ P(A^{\prime} \cap B) = P(B) - P(A \cap B) = 0.3 - 0.2 = 0.1 \] **c)** Find \( P(A \cap B^{\prime}) \): \[ P(A \cap B^{\prime}) = P(A) - P(A \cap B) = 0.5 - 0.2 = 0.3 \] **d)** Find \( P(A^{\prime} \cap B^{\prime}) \): \[ P(A^{\prime} \cap B^{\prime}) = 1 - P(A \cup B) = 1 - 0.6 = 0.4 \] ### 5. Events \( C \) and \( D \) Given: \[ P(C) = 0.1, \quad P(D) = 0.2, \quad P(C \cup D) = 0.3 \] **a)** Determine if \( C \) and \( D \) are mutually exclusive: - If they were mutually exclusive, \( P(C \cup D) = P(C) + P(D) \). \[ P(C) + P(D) = 0.1 + 0.2 = 0.3 \quad \text{(equal to } P(C \cup D)\text{)} \] Thus, \( C \) and \( D \) are mutually exclusive. **b)** Find \( P(C^{\prime}) \) and \( P(C^{\prime} \cap D^{\prime}) \): \[ P(C^{\prime}) = 1 - P(C) = 1 - 0.1 = 0.9 \] \[ P(C^{\prime}