Basic Probability | Mathematics for GCSE/IGCSE - Year 11 PDF Download

Basic Probability

What is probability?

• Definition: Probability refers to the likelihood of an event occurring.
• In Real Life: In everyday scenarios, we use terms like impossible, unlikely, or certain to express probabilities.
• Probability Scale: Mathematically, probabilities are represented on a scale from 0 (impossible) to 1 (certain).
  • Representation: Probabilities can be conveyed as fractions, decimals, or percentages.

What key words and terminology are used with probability?

• An experiment is a task that can be done repeatedly, producing observable or recordable results.
  • Trials are the individual repetitions of an experiment.
• An outcome refers to any possible result of a single trial.
• An event is either a single outcome or a group of outcomes.
  • Events are typically labeled with capital letters like A, B, etc.
  • n(A) represents the number of outcomes belonging to event A.
  • Events can comprise one or multiple outcomes.
• The sample space is the complete set of all potential outcomes from an experiment, often depicted as a list or a table.
  • The probability of event A is symbolized as P(A).

How do I calculate basic probabilities?

• If all outcomes have an equal chance of occurring, then each outcome has the same probability.
  • The probability for each outcome is
  • For instance, if there are 50 marbles in a bag, the probability of selecting a specific marble is 1/50.
• The theoretical probability of an event can be determined without conducting an experiment by dividing the number of outcomes favorable to the event by the total number of outcomes.
  
  • For example, if there are 50 marbles in a bag and 20 of them are blue, then the probability of selecting a blue marble is 20/50.
  • Sometimes, listing or tabulating all possible outcomes can aid in identifying them in certain situations.

How do I find missing probabilities?

• The sum of the probabilities of all outcomes in an experiment equals 1.
  • If you have a probability table with one probability missing, you can find it by subtracting the sum of the other probabilities from 1.
• The complement of event A represents the event where A does not occur, often denoted as A'.
  • This can be conceptualized as the opposite of A.
  • Mathematically, it's represented as P(A) + P(A') = 1.
  • This is commonly expressed as P(A') = 1 - P(A).

What are mutually exclusive events?

• Two events are mutually exclusive if they cannot both occur simultaneously.
  • For instance, in rolling a die, the events "rolling a prime number" and "rolling a 6" are mutually exclusive.
• If events A and B are mutually exclusive, then to calculate the probability of either A or B occurring, you can just add the probabilities of A and B together.
• Complementary events are also mutually exclusive.

Possibility Diagrams

What is a possibility diagram (sample space)?

• In probability, the sample space refers to all the potential outcomes.
• In simple scenarios, the sample space can be presented as a list.
  • For instance, when flipping a coin, the sample space includes "Heads" and "Tails," often abbreviated as H and T.
  • Similarly, for rolling a six-sided die, the sample space comprises the numbers 1 through 6.
• When dealing with combined events, such as rolling two dice and adding their scores, a grid or possibility diagram can be utilized.
  • A comprehensive list of all possibilities would be lengthy and prone to omissions.
  • Identifying missed possibilities or patterns in the sample space would be challenging with a long list.
• Therefore, using a possibility diagram is preferable for clarity and ease of analysis.

• If you need to combine more than two things you'll probably need to go back to listing
  • For example, flipping three coins (or flipping one coin three times!)
    • In this case the sample space is:  HHH, HHT, HTH, THH, HTT, THT, TTH, TTT (8 possible outcomes)

How do I use a possibility diagram to calculate probabilities?

Probabilities can be determined by counting desired outcomes and dividing by the total possibilities in the sample space.

Basic Concept of Probabilities:

• In a sample space like 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, where 4 numbers (2, 3, 5, and 7) are prime, the probability of selecting a prime number is calculated as the number of prime numbers divided by the total numbers in the sample space.
• When rolling two dice and summing the results, for instance, to find the probability of getting an 8, we can observe that there are 5 ways to achieve this outcome out of 36 total possibilities.

Consideration of Fairness:

• It's essential to note that this counting method works effectively when all outcomes in the sample space are equally likely. For example, in a fair six-sided die, each number (1, 2, 3, 4, 5, 6) has an equal likelihood of occurring.
• Similarly, in the case of a fair coin with outcomes 'H' (Heads) and 'T' (Tails), both are equally likely.
• However, when it comes to winning the lottery, the outcomes 'Yes' and 'No' are not equally likely, so the simple counting method cannot be applied here.

Conditional Probability:

• Conditional probability involves finding the likelihood of an event given that another event has occurred. For example, when rolling two dice, we can calculate the probability of one die showing a 6, given that the total sum on both dice is 7.
  • To achieve this, we count the total outcomes that sum to 7 (6 outcomes) as the denominator.
  • Among these, we count the outcomes where one die shows a 6 (2 outcomes: (1, 6) and (6, 1)) as the numerator to find the conditional probability.
  • So the probability is 2/6 = (1/3)