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) $$ P(E \cap F) $$

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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} 1. **Probability of Outcomes from a Single Die Toss:** - **a)** $$ P(4) = \frac{1}{6} $$ - **b)** $$ P(\text{odd}) = \frac{1}{2} $$ - **c)** $$ P(\text{even}) = \frac{1}{2} $$ - **d)** $$ P(<4) = \frac{1}{2} $$ - **e)** $$ P(>4) = \frac{1}{3} $$ - **f)** $$ P(\text{odd or even}) = 1 $$ 2. **Probability in a Junior School Class:** - **a)** $$ P(B \text{ only}) = \frac{3}{28} $$ - **b)** $$ P(T \cup B) = \frac{19}{28} $$ - **c)** $$ P(\text{neither}) = \frac{9}{28} $$ 3. **Probability from a Set of Integers:** - **a)** $$ P(\text{even or divisible by 3}) = \frac{2}{3} $$ - **b)** $$ P(\text{even and not divisible by 3}) = \frac{1}{3} $$ - **c)** $$ P(\text{not even and not divisible by 3}) = \frac{1}{3} $$ 4. **Given Probabilities:** - **a)** $$ P(B) = 0.3 $$ - **b)** $$ P(A^{\prime} \cap B) = 0.1 $$ - **c)** $$ P(A \cap B^{\prime}) = 0.3 $$ - **d)** $$ P(A^{\prime} \cap B^{\prime}) = 0.4 $$ 5. **Events $$ C $$ and $$ D $$:** - **a)** $$ C $$ and $$ D $$ are mutually exclusive. - **b)** $$ P(C^{\prime}) = 0.9 $$, $$ P(C^{\prime} \cap D^{\prime}) = 0.7 $$ 6. **Mutually Exclusive Events $$ A, B, C, D $$:** - **a)** $$ P(E \cup F) = 0.6 $$ - **b)** $$ P(E \cap F) = 0 $$

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