Basic Probability

• Theoretical probability is calculated based on reasoning about the possible outcomes of an experiment, assuming all outcomes are equally likely. It does not require conducting the experiment itself.

• For an event A, its probability, denoted as $P(A)$, is determined by the ratio of the number of favorable outcomes for event A to the total number of possible outcomes in the sample space. This formula is applicable only when each outcome has an equal chance of occurring.

$P(A) = \frac{\text{Number of outcomes favorable to A}}{\text{Total number of possible outcomes}}$

• For example, if a bag contains 5 red balls and 3 blue balls, the total number of outcomes is 8. The probability of drawing a red ball, $P(\text{Red})$, would be $\frac{5}{8}$, as there are 5 favorable outcomes (red balls).

• Sum of Probabilities: The probabilities of all possible individual outcomes in a sample space must sum to 1. This rule is crucial for verifying probability distributions and finding missing probabilities.

• If you are given a set of probabilities for all outcomes except one, you can find the missing probability by subtracting the sum of the known probabilities from 1. This ensures that the entire sample space is accounted for.

• Complement Rule: The complement of an event A, denoted as $A'$ or $A^c$, represents the event that A does not occur. The probability of the complement is found by subtracting the probability of A from 1.

$P(A') = 1 - P(A)$

• This rule is particularly useful when it's easier to calculate the probability of an event happening than the probability of it not happening. For instance, if the probability of rain is 0.3, the probability of no rain is $1 - 0.3 = 0.7$.

• Mutually exclusive events are events that cannot occur at the same time during a single trial of an experiment. If one event happens, the other cannot.

• For example, when rolling a single die, the event 'rolling an even number' and the event 'rolling an odd number' are mutually exclusive because a single roll cannot be both even and odd simultaneously.

• Addition Rule for Mutually Exclusive Events: If two events, A and B, are mutually exclusive, the probability that either A or B occurs is the sum of their individual probabilities. This is often stated as $P(A \text{ or } B) = P(A) + P(B)$.

• Complementary events are a special case of mutually exclusive events where the two events together cover the entire sample space. For example, 'rolling a 6' and 'not rolling a 6' are complementary and thus mutually exclusive.

• Outcome vs. Event: An outcome is a single, specific result of an experiment (e.g., rolling a 3 on a die). An event can be a single outcome or a collection of multiple outcomes (e.g., rolling an odd number, which includes 1, 3, and 5).

• Fair vs. Biased Events: A fair event implies that all individual outcomes in the sample space have an equal probability of occurring (e.g., a fair coin has $P(H) = P(T) = 0.5$). A biased event means that outcomes do not have equal probabilities, often due to an uneven distribution or manipulation (e.g., a loaded die).

• Mutually Exclusive vs. Independent Events: Mutually exclusive events cannot happen at the same time (e.g., rolling a 1 and rolling a 2 on a single die). Independent events are events where the occurrence of one does not affect the probability of the other (e.g., rolling a 6 on the first die and rolling a 6 on the second die). These are distinct concepts and should not be confused.

• Fractions for Precision: Unless specified otherwise, express probabilities as simplified fractions. Fractions maintain exact values, avoiding rounding errors that can occur with decimals or percentages.

• Verify Sum to One: Always check that the probabilities of all possible outcomes in a sample space sum to 1. This is a quick way to identify if any probabilities are missing or incorrectly calculated.

• Identify Event Type: Before applying formulas, determine if events are mutually exclusive. If they are, the addition rule $P(A \text{ or } B) = P(A) + P(B)$ can be used; otherwise, a more general rule might be needed (though not covered in basic probability).

• Clearly Define Sample Space: For any probability problem, explicitly listing or clearly defining the sample space helps ensure that all possible outcomes are considered and none are missed, which is crucial for accurate calculations.

• Assuming Equal Likelihood: A common mistake is to assume all outcomes are equally likely without verification. This assumption is only valid for fair experiments (e.g., fair dice, fair coins) and can lead to incorrect probability calculations if the experiment is biased.

• Incorrectly Defining the Sample Space: Failing to list all possible outcomes or including impossible outcomes in the sample space will lead to incorrect total outcomes, thus skewing probability calculations. Always ensure the sample space is comprehensive and accurate.

• Confusing 'And' with 'Or': Students often mix up the conditions for 'A and B' versus 'A or B'. For basic probability, 'A or B' for mutually exclusive events means adding probabilities, while 'A and B' for independent events involves multiplication (a concept typically introduced after basic probability).

• Misinterpreting the Complement Rule: While $P(A') = 1 - P(A)$ is straightforward, errors can occur if $P(A)$ itself is miscalculated or if the event $A$ is not clearly defined. Always ensure you are finding the complement of the intended event.