a) i) Probability is a numerical measure of the likelihood that an event will occur. It is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.

Here's the breakdown of the probability questions: a) i) Probability is a numerical measure of the likelihood that an event will occur. It is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. a) ii) The Three Axioms of Probability (Kolmogorov Axioms) are: 1. Non-negativity: For any event E, the probability of E is non-negative. P(E) 0 2. Normalization: The probability of the sample space S (the set of all possible outcomes) is 1. P(S) = 1 3. Additivity (for mutually exclusive events): For any sequence of mutually exclusive events E_1, E_2, E_3, , the probability of their union is the sum of their individual probabilities. P(E_1 E_2 E_3 ) = P(E_1) + P(E_2) + P(E_3) + b) i) Step 1: Determine the total number of presidents and military personnel. Total presidents = 14 Military personnel = 8 Step 2: Calculate the number of civilian presidents. Civilian presidents = Total presidents - Military personnel Civilian presidents = 14 - 8 = 6 Step 3: Calculate the probability of randomly selecting a civilian president. P(civilian) = Number of civilian presidentsTotal number of presidents P(civilian) = (6)/(14) = (3)/(7) The probability of selecting a president who was a civilian is (3)/(7). b) ii) • Experiment: A procedure that can be repeated, has a well-defined set of possible outcomes, and whose outcome is not known in advance. • Sample Space: The set of all possible outcomes of a random experiment. It is usually denoted by S. • Event Space: A collection of possible outcomes of an experiment. It is a subset of the sample space. • Sample Point: A single possible outcome of a random experiment. It is an element of the sample space. • Trials: A single performance of an experiment. c) When a fair coin is tossed twice, the sample space S is: S = \HH, HT, TH, TT\ The total number of possible outcomes is 4. c) i) at least one head show up Step 1: Identify outcomes with at least one head. These outcomes are \HH, HT, TH\. Number of favorable outcomes = 3. Step 2: Calculate the probability. P(at least one head) = Number of favorable outcomesTotal number of outcomes = (3)/(4) The probability of at least one head is (3)/(4). c) ii) only tails show up Step 1: Identify outcomes with only tails. This outcome is \TT\. Number of favorable outcomes = 1. Step 2: Calculate the probability. P(only tails) = Number of favorable outcomesTotal number of outcomes = (1)/(4) The probability of only tails is (1)/(4). c) iii) different side show up Step 1: Identify outcomes where different sides show up (one head and one tail). These outcomes are \HT, TH\. Number of favorable outcomes = 2. Step 2: Calculate the probability. P(different sides) = Number of favorable outcomesTotal number of outcomes = (2)/(4) = (1)/(2) The probability of different sides showing up is (1)/(2). That's 2 down. 3 left today — send the next one.