Counting Outcomes

Understanding Counting Outcomes

  • Counting outcomes is a part of probability theory that deals with the determining the number of possible outcomes in an event.
  • An outcome is the result of a single trial in an experiment.
  • An experiment refers to a process (like tossing a coin or rolling a die) where the outcomes are uncertain.
  • The sample space of an experiment is the set of all possible outcomes.

Counting Techniques

  • When all outcomes are equally likely, the probability of an event happening is calculated by dividing the number of favourable outcomes by the total number of outcomes.
  • The fundamental principle of counting: In a multi-stage event, the total number of outcomes is equal to the product of the number of outcomes at each stage.
  • For example, if a coin is tossed twice, the number of outcomes is 2 (head or tail at first toss) times 2 (head or tail at the second toss), which equals 4. So, the outcomes can be HH, HT, TH, TT.
  • A tree diagram can help visualise multi-stage events and calculate the possible outcomes.
  • Permutations count outcomes when the order matters, and combinations count outcomes when the order does not matter.

Using Counting in Probability

  • To calculate probability using the counting techniques, decide whether order is important (permutation) or irrelevant (combination).
  • If a die is rolled and a coin is tossed, there are 6 possibilities for the die roll and 2 for the coin toss. So, the total number of outcomes is 6 times 2, equals 12.
  • For combinations, use the formula nCr = n! / [r!(n-r)!], where n is the number of total elements, r is the number of elements to choose, ‘!’ denotes factorial (eg. 3! = 3 x 2 x 1).
  • For permutations, use the formula nPr = n! / (n-r)!.

Handling Complex Outcomes

  • For counting outcomes in complex scenarios, breaking down the problem into simpler events often helps.
  • Remember, the probability of an impossible event is 0 and the probability of a certain event is 1.
  • For compound events, if the events are mutually exclusive (cannot happen together), add the probabilities. If the events are not mutually exclusive (can happen together), the probability of either event happening is the sum of the probabilities minus the probability of both events happening.

Reviewing Counting Principles

  • In conclusion, counting principles are essential for calculating probabilities.
  • Understand the difference between permutations and combinations, and when to use each.
  • Practice a range of problems to become familiar with the conditions and calculations involved.