Sample Space
The sample space, often symbolised by S, signifies all conceivable outcomes of a random experiment. It's essentially a set that encompasses every potential result.
- Definition: The sample space for a set of events is the collection of all possible values the events might assume. Formally, the set of feasible events for a given random variable forms a sigma-algebra, and the sample space is defined as the most extensive set in this sigma-algebra. This space can also be referred to as an event space or possibility space. Understanding the sample space is essential when working with Venn Diagrams.
- Example: Ponder upon tossing a fair coin. The sample space is S = {Heads, Tails} since these are the sole two possible outcomes.
- Example: Rolling a standard six-sided die would result in a sample space of S = {1, 2, 3, 4, 5, 6}.
- Example: For the toss of two coins, each of which may land heads (H) or tails (T), the sample space encompasses all possible outcomes: HH, HT, TH, and TT.
Events
An event represents a specific outcome or a combination of outcomes derived from the sample space. Essentially, it's a subset of the sample space.
- Example: In the coin toss experiment, obtaining a "Heads" is an event.
- Example: In the die-rolling experiment, achieving an even number is an event, symbolised as E = {2, 4, 6}. Events can also be analysed using the principles of the Binomial Distribution.
Probability Axioms
Probability axioms are the bedrock principles that delineate the probability of events. They are:
1. Non-negativity: The probability of any event E is invariably non-negative. Mathematically, P(E) is greater than or equal to 0.
2. Certainty: The probability of the entire sample space is 1. That is, P(S) = 1.
3. Additivity: If two events A and B are mutually exclusive (they can't transpire simultaneously), then the probability of either A or B occurring is the summation of their individual probabilities. Mathematically, P(A union B) = P(A) + P(B) for mutually exclusive events. To explore more about these concepts, consider looking into Bayes' Theorem.
Example Question:
Suppose you extract a card from a standard deck of 52 playing cards. What's the probability that the card is a red queen?
Solution: There are 2 red queens in a standard deck (Queen of Hearts and Queen of Diamonds). Thus, the probability is: P(Red Queen) = Number of favourable outcomes / Total outcomes in sample space P(Red Queen) = 2/52 = 1/26
Delving Deeper into Probability
In colloquial parlance, "probability" denotes the likelihood or chance of a particular event transpiring. It's expressed on a linear scale ranging from 0 (indicating impossibility) to 1 (indicating certainty). This can also be represented as a percentage, spanning from 0% (no chance) to 100% (absolute certainty). Probabilities are often visualised using scatter plots to show relationships between variables.
Example: If the weather forecast suggests a 90% chance of rain, it implies that out of 100 similar days, we'd anticipate rain on 90 of them.
When working with larger sets of data, the normal distribution is a key concept to understand how probabilities are distributed across different outcomes.
FAQ
No, an event cannot have a probability greater than 1 or less than 0. The probability of any event is always bounded between 0 and 1, inclusive. A probability of 0 indicates that the event is impossible and will not occur, while a probability of 1 indicates that the event is certain and will always occur. Any value outside this range would violate the basic axioms and principles of probability. Assigning a probability greater than 1 or less than 0 would not make logical or mathematical sense and would lead to inconsistencies in probabilistic reasoning.
The probability of the entire sample space being 1 is a fundamental axiom of probability. It stems from the idea that the sample space encompasses all possible outcomes of a random experiment. Since one of the outcomes in the sample space must occur when the experiment is conducted, the total probability distributed across all these outcomes must sum up to 1. In other words, the certainty that some outcome will occur is 100%, or a probability of 1. This foundational concept ensures that probabilities across all individual events in the sample space don't exceed the realm of possibility.
Two events are said to be mutually exclusive if they cannot occur at the same time. In other words, the occurrence of one event automatically implies the non-occurrence of the other. For instance, when tossing a fair coin, the events "getting a Heads" and "getting a Tails" are mutually exclusive because both cannot happen simultaneously. If one event occurs, the other cannot. Mathematically, for two mutually exclusive events A and B, the intersection of A and B is an empty set, meaning P(A and B) = 0. This concept is essential in probability theory as it helps in determining the combined probability of multiple events.
Probability and odds are both measures of the likelihood of an event occurring, but they are expressed differently. Probability is the ratio of the number of favourable outcomes to the total number of possible outcomes. It's a value between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. Odds, on the other hand, represent the ratio of favourable outcomes to unfavourable outcomes. For instance, if the probability of an event occurring is 1/4, the odds in favour of the event are 1 to 3, often written as 1:3. Essentially, while probability gives the likelihood of an event in relation to all possible outcomes, odds compare the likelihood of the event occurring versus it not occurring.
The terms "sample space" and "event space" are closely related but have distinct meanings in probability theory. The sample space, often denoted by S, represents the set of all possible outcomes of a random experiment. It encompasses every conceivable result of that experiment. On the other hand, an event space refers to a collection of specific outcomes or combinations of outcomes from the sample space. In other words, while the sample space lists all potential outcomes, the event space lists subsets of these outcomes that we are particularly interested in. For instance, in a coin toss, the sample space is {Heads, Tails}, but an event could be getting a "Heads", which is a subset of the sample space.
Practice Questions
When two balls are drawn without replacement, the possible combinations of colours are:
- Red and Red (RR)
- Red and Green (RG)
- Red and Blue (RB)
- Green and Red (GR)
- Green and Green (GG)
- Green and Blue (GB)
- Blue and Red (BR)
- Blue and Green (BG)
- Blue and Blue (BB)
Thus, the sample space for the colours of the two balls drawn is: {RR, RG, RB, GR, GG, GB, BR, BG, BB}.
To determine the probability, we first need to identify the combinations that result in a sum of 7. These combinations are:
- 1 and 6 (1,6)
- 2 and 5 (2,5)
- 3 and 4 (3,4)
- 4 and 3 (4,3)
- 5 and 2 (5,2)
- 6 and 1 (6,1)
There are 6 favourable outcomes. Since there are 6 possible outcomes for the first roll and 6 for the second roll, there are a total of 6 x 6 = 36 possible outcomes when rolling the die twice.
Thus, the probability that the sum is 7 is given by the ratio of favourable outcomes to total outcomes: 6/36 = 1/6.
Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.