Expected Frequency

Statement

The expected frequency tells us how many times an event is likely to occur in a set of trials, assuming the probability of success remains constant. It is a key tool in statistics and probability, particularly for predicting outcomes and for comparing observed and theoretical results. The formula is:

\[ E = n p \]

Where:

• \(E\) = expected frequency
• \(n\) = total number of trials
• \(p\) = probability of success in each trial

Why it works

• In probability, if the chance of an event is \(p\), then in one trial the expected count is \(p\).
• Over \(n\) trials, the expected number of successes is \(n \times p\).
• This is the mean of the binomial distribution \(B(n, p)\).

Recipe (Steps to Apply)

1. Identify \(n\), the number of trials or sample size.
2. Identify \(p\), the probability of success (in decimal or fraction form).
3. Multiply: \(E = n \times p\).
4. Interpret the result: it is the expected (average) number of successes if the experiment were repeated many times.

Spotting it

Use this formula whenever you are asked to find an expected count in probability or statistics, such as:

• How many sixes in 120 dice rolls?
• How many heads in 50 coin tosses?
• How many students are expected to pass if the probability of passing is 0.7 and there are 40 students?

Common pairings

• Relative frequency: observed frequency ÷ total trials.
• Chi-squared tests: comparing observed frequency with expected frequency.
• Binomial distribution: expected value = \(np\).

Mini examples

1. Given: Toss a fair coin 100 times. Find: Expected heads. Answer: 50.
2. Given: Roll a die 60 times. Find: Expected sixes. Answer: 10.

Pitfalls

• Forgetting to convert percentages to decimals: Always use \(p\) as a decimal or fraction (e.g. 25% = 0.25).
• Misinterpreting expected frequency: It is not a guarantee, but an average outcome across many repetitions.
• Confusing \(n\) with observed data: Only the planned number of trials goes into the formula.

Exam Strategy

• Write \(E = np\) at the start of your working.
• Check units: if probabilities are in percentages, convert them.
• Round to 1 decimal place if asked for an approximation.
• Interpret results clearly: “We expect about 7 students…”

Worked Micro Example

A factory makes light bulbs. The probability that a bulb is faulty is 0.02. If 500 bulbs are made, what is the expected number of faulty bulbs?

Here, \(n = 500\), \(p = 0.02\).

\(E = 500 \times 0.02 = 10\).

So, we expect about 10 faulty bulbs.

Summary

The expected frequency formula is one of the simplest yet most powerful tools in statistics. It provides a way of predicting average results over the long run. Whether in coins, dice, surveys, or quality control, it gives a baseline expectation that can be compared with observed data to test fairness, accuracy, or randomness.