Planning & Decision Making · Topic 6 of 11
Calculating Basic Probabilities
Theory
Probability measures the likelihood or chance of a specific outcome happening. We can describe probability using words like 'certain', 'likely', or 'impossible', but in mathematics, we measure it precisely as a fraction, decimal, or percentage.
1. Definitions and Terminology
To discuss probability accurately, you must understand the correct terminology:
- Trial: The probability experiment you are looking at (e.g., rolling a dice, or selecting a person at random).
- Outcomes: All the possible results of the probability experiment. This is also referred to as the sample space.
- Event: The particular outcome you are interested in. This is also known as a "success".
- Notation: We use the letter P followed by brackets to denote probability. For example, P(Heads) means "the probability of the event of flipping a coin and it landing on heads".
2. Calculating Single Events
To calculate the basic probability of an event occurring, we use the following formula:
- The Probability Scale: Probabilities must always be between 0 and 1. If , the event is absolutely certain to happen. If , the event is impossible. A probability can never be a negative number or greater than 1.
- The "NOT" Rule: When we need to calculate the probability that something does not happen, we subtract the probability that it does happen from 1. Formula: .
3. Types of Events
You must be able to classify how different events relate to each other:
- Independent Events: Two events are independent if they do not affect each other. For example, rolling a dice twice; getting a 6 on the first roll does not lessen the chance of getting a 6 on the next roll.
- Dependent Events: This means the outcome is affected by the previous event. For example, drawing a card from a deck and not putting it back lowers the total number of cards for the next draw, changing the probabilities.
- Mutually Exclusive Events: Two outcomes are mutually exclusive if they cannot occur at the same time. For example, a single coin flip cannot result in both heads and tails simultaneously.
4. The Gambler's Fallacy (Monte Carlo Fallacy)
This is the incorrect belief that if a particular independent event occurs more frequently than normal in the past, it is "due" to happen less frequently in the future (or vice versa).
Example: If a roulette wheel lands on black 26 times in a row, a gambler might incorrectly assume the next spin is highly likely to be red. In reality, the events are completely independent, and the chance of red remains unchanged.
Worked examples
Example 1
Example 1: Calculating Basic Probabilities and the 'NOT' Rule
A bowl of fruit contains 5 apples, 4 bananas, and 3 oranges. A student selects one piece of fruit at random.
(a) Calculate the probability that the student selects a banana.
(b) Calculate the probability that the student does not select an apple.
(a)
Total number of fruit = . Number of bananas = 4.
(or 0.333... or 33.3%).
(b)
First, find the probability of picking an apple: .
Then use the NOT rule: .
(Alternatively, simply add the bananas and oranges together: ).
Example 2
Example 2: Identifying Types of Events
State whether the following pairs of events are mutually exclusive, independent, or dependent. Justify your answers.
(a) Rolling an even number on a standard die, and rolling an odd number on the same die.
(b) Flipping a 'Heads' on a coin, and rolling a '6' on a die.
(c) Drawing a card from a deck and keeping it, then drawing a second card from the same deck.
(a)
Mutually exclusive. You cannot roll a number that is both even and odd at the exact same time.
(b)
Independent. The outcome of the coin flip has absolutely no physical effect on the outcome of the die roll.
(c)
Dependent. Because the first card is kept and not replaced, the total number of cards in the deck changes, which directly affects the probability of the second draw.
Example 3
Example 3: The Gambler's Fallacy
A student is playing a game where they flip a fair coin. The coin lands on 'Heads' 5 times in a row. The student tells their friend, "I am definitely going to bet on Tails for the next flip, because it is well overdue."
State the name of the statistical fallacy the student is demonstrating, and explain why their logic is incorrect.
The student is demonstrating the Gambler's Fallacy (or the Monte Carlo Fallacy).
Their logic is completely incorrect because every single coin flip is an independent event. The previous 5 flips have absolutely no physical effect on the 6th flip, so the probability of getting 'Tails' remains exactly 0.5 (or 50%), regardless of what happened before.