Probability

Below you will find the summaries of probability as found in the Via Afrika study guides.

What is Probability?

Probability is a way of measuring how likely something is to happen. It's expressed as a number between 0 and 1 (or as a percentage between 0% and 100%).

• 0 (or 0%): Means the event is impossible. It will definitely not happen.

  • Example: The probability of the sun rising in the west tomorrow is 0.

• 1 (or 100%): Means the event is certain. It will definitely happen.

  • Example: The probability of a 6-sided die showing a number less than 7 when rolled is 1.

• 0.5 (or 50%): Means there's an even chance of the event happening or not happening.

  • Example: The probability of getting "Heads" when you toss a fair coin is 0.5.

Key Terms You'll Encounter:

1. Experiment: This is an action or a test where the outcome isn't known beforehand.

   • Example: Tossing a coin, rolling a dice, drawing a card from a shuffled deck.

2. Outcome: This is one of the possible results of an experiment.

   • Example: If you toss a coin, the outcomes are "Heads" or "Tails". If you roll a standard 6-sided die, the outcomes are 1, 2, 3, 4, 5, or 6.

3. Sample Space (S): This is the list of all the possible outcomes of an experiment.

   • Example: For tossing a coin, S = {Heads, Tails}. For rolling a die, S = {1, 2, 3, 4, 5, 6}.

4. Event: This is a specific outcome or a group of outcomes that you are interested in.

   • Example: When rolling a die, "getting an even number" is an event (outcomes: {2, 4, 6}). "Getting a 5" is also an event (outcome: {5}).

How to Calculate Probability (The Basic Formula):

The most common way to calculate the probability of an event is:

P(Event) = Number of favourable outcomes ÷ Total possible outcomes

• Favourable outcomes: These are the outcomes that fit the event you're looking for.

• Total possible outcomes: This is the size of your sample space.

Example: What is the probability of rolling an even number on a standard 6-sided die?

• Experiment: Rolling a die.

• Sample Space (S): {1, 2, 3, 4, 5, 6} (Total possible outcomes = 6)

• Event: Getting an even number.

• Favourable outcomes: {2, 4, 6} (Number of favourable outcomes = 3)

P(Even number) = 3 ÷ 6 ​ = ½ ​= 0.5 or 50%

 Types of Probability:

1. Theoretical Probability: This is what we expect to happen based on how the event is set up (like a fair coin or a balanced die). We calculate it without actually doing the experiment.

   • Example: The theoretical probability of getting "Heads" is 0.5, even if you haven't tossed a coin yet.

2. Experimental Probability (or Relative Frequency): This is what actually happens when you perform an experiment many times. It's calculated based on the results you observe.

   • P(Event)=Total number of trials (how many times you did the experiment)Number of times the event occurred​

   • Example: If you toss a coin 100 times and get 48 Heads, the experimental probability of getting Heads is 10048​=0.48. You'll notice that as you do more and more trials, your experimental probability usually gets closer to the theoretical probability.

Using Tools to Understand Probability:

• Tree Diagrams: These are visual tools that use "branches" to show all possible outcomes of a sequence of events. They are great for visualizing sample spaces for multiple events.

• Two-Way Tables (Contingency Tables): These tables help organize data for two different categories, making it easy to calculate probabilities involving both categories.

  • Example: A table showing how many boys and girls in a class like soccer or netball. From this, you can figure out the probability of a student being a boy AND liking soccer.

Why is Probability Important in Maths Literacy?

In Maths Literacy, understanding probability helps you:

• Make informed decisions: Should you take an umbrella based on the weather forecast's probability of rain?

• Interpret information: What does it mean if a medical test has a 95% accuracy rate?

• Assess risk: How risky is it to invest in a certain stock based on past performance probabilities?

• Understand games of chance: Why are casino games designed so the house usually wins?

In essence, probability in Maths Literacy is about being smart about uncertainty and using numbers to give yourself a better idea of what might happen