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Revision

Chapter 10: Probability

  • This chapter provides good opportunity for experiments and activities in classroom where you can illustrate theoretical probability and number of possible arrangements in practice. Real-life examples have been used extensively in the exercise sections and you may choose to illustrate some of these concepts experimentally in class.
  • The terminology and usage of language in this section can be confusing, especially to second-language speakers. Discuss terminology regularly and emphasise the careful reading of questions.
  • Union and intersection symbols have been included, but “and” and “or” is the preferred notation in CAPS.
  • Make sure to outline the differences between 'and', 'or', 'only' and 'both'. For example, there may be no difference between tea and coffee drinkers and tea or coffee drinkers in common speech but in probability, the 'and' and 'or' have very specific meanings. Tea and coffee drinkers refers to the intersection of tea drinkers, i.e. those who drink both beverages, while tea or coffee drinkers refers to the union, i.e. those who drink only tea, those who drink only coffee and those who drink both.
  • Some learners may find factorial notation challenging. A lengthy problem-set has been provided to try and iron out some common misconceptions, such as \(4!3! \ne 12!\) or \(\frac{6!}{4!} \ne \frac{3!}{2!}\), but you may have to go over this more slowly with some learners.
  • Note that the formula for the arrangement of \(n\) different items in \(r\) different places i.e. \(\frac{n!}{(n-r)!}\) is NOT included in CAPS and learners should therefore be able to solve these problems logically.
  • When applying the fundamental counting principle to probability problems, learners may struggle with knowing when to multiply and when to add probabilities. When a number of different outcomes fit a desired result, the probabilities of each outcome are added. When determining the probability of two or more events occurring, their individual probabilities are multiplied.

10.1 Revision (EMCJV)

Terminology (EMCJW)

Outcome: a single observation of an uncertain or random process (called an experiment). For example, when you accidentally drop a book, it might fall on its cover, on its back or on its side. Each of these options is a possible outcome.

Sample space of an experiment: the set of all possible outcomes of the experiment. For example, the sample space when you roll a single \(\text{6}\)-sided die is the set \(\left\{1;2;3;4;5;6\right\}\). For a given experiment, there is exactly one sample space. The sample space is denoted by the letter \(S\).

Event: a set of outcomes of an experiment. For example, if you have a standard deck of 52 cards, an event may be picking a spade card or a king card.

Probability of an event: a real number between and inclusive of \(\text{0}\) and \(\text{1}\) that describes how likely it is that the event will occur. A probability of \(\text{0}\) means the outcome of the experiment will never be in the event set. A probability of \(\text{1}\) means the outcome of the experiment will always be in the event set. When all possible outcomes of an experiment have equal chance of occurring, the probability of an event is the number of outcomes in the event set as a fraction of the number of outcomes in the sample space. To calculate a probability, you divide the number of favourable outcomes by the total number of possible outcomes.

Relative frequency of an event: the number of times that the event occurs during experimental trials, divided by the total number of trials conducted. For example, if we flip a coin \(\text{10}\) times and it landed on heads \(\text{3}\) times, then the relative frequency of the heads event is \(\frac{3}{10} = \text{0,3}\).

Union of events: the set of all outcomes that occur in at least one of the events. For \(\text{2}\) events called \(A\) and \(B\), we write the union as “\(A \text{ or } B\)”. Another way of writing the union is using set notation: \(A \cup B\). For example, if \(A\) is all the countries in Africa and \(B\) is all the countries in Europe, \(A\) or \(B\) is all the countries in Africa and Europe.

Intersection of events: the set of all outcomes that occur in all of the events. For \(\text{2}\) events called \(A\) and \(B\), we write the intersection as “\(A \text{ and } B\)”. Another way of writing the intersection is using set notation: \(A \cap B\). For example, if \(A\) is soccer players and \(B\) is cricket players, \(A\) and \(B\) refers to those who play both soccer and cricket.

Mutually exclusive events: events with no outcomes in common, that is \((A \text{ and } B)\) is an empty set. Mutually exclusive events can never occur simultaneously. For example the event that a number is even and the event that the same number is odd are mutually exclusive, since a number can never be both even and odd.

Complementary events: two mutually exclusive events that together contain all the outcomes in the sample space. For an event called \(A\), we write the complement as “\(\text{not } A\)”. Another way of writing the complement is as \(A'\).

Dependent and independent events: two events, \(A\) and \(B\), are independent if the outcome of the first event does not influence the outcome of the second event. For example, if you flip a coin and it lands on tails and flip it again and it lands on heads, neither outcome influences the other. Two events, C and D, are dependent if the outcome of one event influences the outcome of the other. For example, if your lunchbox contains \(\text{3}\) sandwiches and \(\text{2}\) apples, when you eat one of the items, this reduces the number of choices you have when deciding to eat a second item.