Welcome to Probability: Understanding Chance and Estimating Likelihood
Have you ever checked the weather forecast to see if it might rain, or wondered about your chances of rolling a six in a board game? If so, you have already been thinking about probability! In this chapter, we will explore how to measure chance using the probability scale and how to calculate and estimate probabilities from theory and real-life data. Don't worry if maths hasn't always been your favourite subject—we will break down every single idea into bite-sized, easy-to-follow steps.
1. The Probability Scale
Probability is a measure of how likely an event is to happen. We can describe probability using everyday words, but in statistics, we give it an exact numerical value between \(0\) and \(1\).
Words vs. Numbers
On the probability scale:
• Impossible has a probability of exactly \(0\) (or \(0\%\)). For example, rolling an \(8\) on a standard six-sided die.
• Unlikely lies between \(0\) and \(0.5\). For example, winning a raffle when you bought \(1\) ticket out of \(100\).
• Evens (or Even chance) has a probability of exactly \(0.5\) (or \(\frac{1}{2}\), or \(50\%\)). For example, getting Heads when flipping a fair coin.
• Likely lies between \(0.5\) and \(1\). For example, pulling a red bead from a bag containing \(90\) red beads and \(10\) blue beads.
• Certain has a probability of exactly \(1\) (or \(100\%\)). For example, the sun rising in the east tomorrow.
How Can We Write Probabilities?
You can write any probability in three acceptable mathematical formats:
1. Fractions: e.g. \(\frac{1}{4}\)
2. Decimals: e.g. \(0.25\)
3. Percentages: e.g. \(25\%\)
Top Tip: A probability can never be less than \(0\) and can never be greater than \(1\) (or \(100\%\)). If your calculation gives you an answer like \(1.4\) or \(-0.2\), pause and double-check your working!
Key Takeaway: All probabilities lie on a scale from \(0\) (impossible) to \(1\) (certain). You can write them as fractions, decimals, or percentages.
2. Theoretical Probability
Theoretical probability is what we expect to happen based on pure mathematics and symmetry, assuming all possible outcomes are equally likely.
The Core Formula
When outcomes are equally likely, we calculate the probability of an event \(A\), written as \(P(A)\), using:
\(P(A) = \frac{\text{Number of successful outcomes}}{\text{Total number of possible outcomes}}\)
Step-by-Step Example: Rolling a Fair Die
Imagine you roll a fair, six-sided die. What is the probability of rolling an even number?
Step 1: Find the total number of possible outcomes.
The possible numbers are \(1, 2, 3, 4, 5, 6\). So, the total number of outcomes is \(6\).
Step 2: Count the successful outcomes (the even numbers).
The even numbers are \(2, 4, 6\). So, there are \(3\) successful outcomes.
Step 3: Put them into the formula.
\(P(\text{Even}) = \frac{3}{6} = \frac{1}{2} = 0.5\)
Did You Know?
The notation \(P(A)\) simply means "the probability of event \(A\) happening". For example, \(P(\text{Heads})\) means "the probability of getting Heads".
Key Takeaway: Theoretical probability is calculated by dividing the number of ways an event can happen by the total number of possible outcomes.
3. The Complement of an Event (Not Happening)
The complement of an event is simply the event not happening. The sum of the probability of an event happening and the probability of it not happening is always equal to \(1\).
\(P(\text{not } A) = 1 - P(A)\)
Example
If the probability that it rains tomorrow is \(0.3\), what is the probability that it does not rain tomorrow?
\(P(\text{no rain}) = 1 - 0.3 = 0.7\)
Key Takeaway: To find the chance of something not happening, subtract the probability of it happening from \(1\).
4. Experimental Probability (Relative Frequency)
Sometimes we cannot use theoretical probability because the outcomes are not equally likely, or the situation is too complex (such as testing the reliability of a machine or checking if a drawing pin lands point up). In these cases, we carry out experiments or gather data. This is called experimental probability or relative frequency.
The Relative Frequency Formula
\(\text{Relative Frequency} = \frac{\text{Frequency of the event}}{\text{Total number of trials}}\)
Step-by-Step Example: Biased Spinner
A student spins a coloured spinner \(50\) times. It lands on Blue \(18\) times. What is the relative frequency of landing on Blue?
Step 1: Identify the frequency of the event (\(18\)).
Step 2: Identify the total number of trials (\(50\)).
Step 3: Calculate relative frequency:
\(\text{Relative Frequency} = \frac{18}{50} = \frac{9}{25} = 0.36\)
Sample Size and Reliability
How reliable is experimental probability? Think about flipping a coin \(4\) times—you might easily get \(4\) Heads just by chance (relative frequency = \(1.0\)). But if you flip the coin \(1000\) times, the relative frequency will get much closer to \(0.5\).
• Small number of trials: The estimate can be unreliable and affected by random chance.
• Large number of trials: As the number of trials increases, the relative frequency gets closer and closer to the true probability. A larger sample size gives a more reliable estimate.
Key Takeaway: Relative frequency is probability based on observed results. Increasing the number of trials makes your estimate much more accurate and reliable.
5. Expected Frequency (Estimating How Many Times)
Once you know the probability of an event, you can predict how many times you expect that event to occur over a given number of trials. This is called the expected frequency.
The Expected Frequency Formula
\(\text{Expected Frequency} = n \times p\)
where \(n\) is the number of trials and \(p\) is the probability of the event occurring.
Step-by-Step Example: Rolling a Die Multiple Times
If you roll a standard fair six-sided die \(300\) times, how many times would you expect to roll a \(5\)?
Step 1: Find the probability of rolling a \(5\):
\(p = \frac{1}{6}\)
Step 2: Identify the number of trials:
\(n = 300\)
Step 3: Multiply \(n\) by \(p\):
\(\text{Expected Frequency} = 300 \times \frac{1}{6} = \frac{300}{6} = 50\)
You would expect to roll a five approximately \(50\) times.
Remember: An expected frequency is an estimate or average. In real life, you might get \(48\) or \(53\) fives due to natural variation!
Key Takeaway: To find the expected number of outcomes, multiply the total number of trials by the probability of the event.
6. Common Mistakes to Avoid in Exams
• Writing probabilities as ratios: Never write a probability as a ratio like \(1:6\). Always use a fraction (\(\frac{1}{6}\)), decimal (\(0.167\)), or percentage (\(16.7\%\)).
• Confusing theoretical and experimental: Theoretical is what should happen by math rules; experimental is what did happen in real trials.
• Thinking past events affect future independent events: If you flip a fair coin and get Heads \(5\) times in a row, the probability of getting Heads on the next flip is still \(\frac{1}{2}\). The coin has no memory!
• Forgetting to simplify or check bounds: Always check that your final answer is between \(0\) and \(1\).
Quick Chapter Summary
• Probability Scale: Runs from \(0\) (impossible) to \(1\) (certain). Evens is \(0.5\).
• Theoretical Probability: \(P(A) = \frac{\text{Successful outcomes}}{\text{Total possible outcomes}}\)
• Complement: \(P(\text{not } A) = 1 - P(A)\)
• Relative Frequency: \(\text{Relative Frequency} = \frac{\text{Frequency}}{\text{Total trials}}\)
• Expected Frequency: \(\text{Expected Frequency} = n \times p\)
• Reliability: More trials = more reliable probability estimates.