4.4.1 Modelling with Distributions

In statistics, we use probability distributions to model real-world situations and predict outcomes. A distribution describes how values of a random variable are spread out. Understanding how to model with distributions is essential for solving problems.

Common Distributions

Binomial Distribution
Normal Distribution

1. Binomial Distribution

Definition: A binomial distribution models the number of successes in a fixed number of independent trials, where each trial has the same probability of success.

When to Use Binomial Distribution:

Fixed number of trials.
Each trial has only two possible outcomes (success or failure).
Trials are independent.
The probability of success remains constant.

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Example: Imagine you have a biased coin that lands on heads 60% of the time. You flip it 10 times. The probability of getting exactly 6 heads is an example of a binomial distribution.

Number of trials (n): 10
Probability of success (p): 0.6
Random variable X: Number of heads in 10 flips. The probability mass function for the binomial distribution is:

P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}

To find $P(X = 6)$ , substitute the values:

P(X = 6) = \binom{10}{6} (0.6)^6 (0.4)^4

This formula calculates the probability of getting exactly 6 heads out of 10 flips.

2. Normal Distribution

Definition: The normal distribution is a continuous probability distribution that is symmetric around the mean. It's often used to model real-world variables like heights, exam scores, and measurement errors, which naturally cluster around a central value.

Key Properties:

Symmetrical bell shape.
Mean (μ) = Median = Mode.
The spread of the distribution is determined by the standard deviation (σ).
About 68% of the data falls within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations (Empirical Rule). When to Use Normal Distribution:
The variable is continuous and can take any value within a range.
The distribution of the data is symmetric and bell-shaped.
You have a large sample size (Central Limit Theorem).

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Example: Suppose the heights of adult males in a town are normally distributed with a mean height of 175 cm and a standard deviation of 10 cm. To find the probability that a randomly selected male is taller than 185 cm, you use the normal distribution.

Step 1: Calculate the z-score

The $z-score$ standardises the height to compare it with the normal distribution.

Z = \frac{X - \mu}{\sigma} = \frac{185 - 175}{10} = 1

Step 2: Use the z-table

The $z-table$ (or calculator) gives the probability that a standard normal variable is less than a particular z-value. For $Z = 1$ , $P(Z < 1)$ is approximately 0.8413.

Step 3: Find the required probability

Since you want the probability that the height is more than 185 cm:

P(X > 185) = 1 - P(Z < 1) = 1 - 0.8413 = 0.1587

This means there is about a 15.87% chance that a randomly selected male is taller than 185 cm.

Modelling Process

Identify the Situation:

Determine which distribution fits the context (e.g., Binomial for a fixed number of trials, Normal for continuous data like height).

Set Parameters:

Define the parameters of the distribution (e.g., mean, standard deviation, probability of success).

Calculate Probabilities:

Use the appropriate formula, statistical tables, or calculators to calculate the required probabilities.

Interpret Results:

Understand the context of the problem to interpret what the probability means in real terms.

Conclusion

Modelling with distributions, particularly the Binomial and Normal distributions, allows us to predict and analyse real-world scenarios effectively. By practising with these distributions, you will develop a strong understanding of how to apply them in different contexts, preparing you for both exam questions and practical applications.

Modelling with Distributions Simplified Revision Notes for A-Level AQA Maths Statistics