Expected Frequencies

If a single trial has probability $p$ of producing a particular event, and the trial is repeated $m$ times independently, then the expected frequency of the event is

$E = m \times p$.

This is exactly $E(X)$ for $X \sim B(m, p)$. The expected frequency is the long-run average count; in any particular sample of $m$ trials the observed count will fluctuate around this value.

Worked through the hook: A fair die rolled $300$ times.

• (a) Probability of rolling a six on one roll: $p = \tfrac{1}{6}$. Expected frequency: $E = 300 \times \tfrac{1}{6} = 50$ sixes.
• (b) Probability of rolling an even number on one roll: $p = \tfrac{1}{2}$. Expected frequency: $E = 300 \times \tfrac{1}{2} = 150$ even rolls.
• Will you get exactly $50$ sixes? Probably not — but typically you'll be within a few of $50$. For $B(300, \tfrac{1}{6})$ the SD is $\sqrt{300 \times \tfrac{1}{6} \times \tfrac{5}{6}} \approx 6.45$, so $50 \pm 6$ is the typical range.

Expected frequency formula: $E = m\times p$ where $m$ is the number of trials and $p$ is the probability of the event on each trial.

Pause — copy the expected frequency formula $E=mp$ and work through the hook example (die rolled 300 times, expected frequency of each face $= 300\times\frac{1}{6}=50$) into your book.

We just saw that if a single trial has probability $p$ and is repeated $m$ times, the expected frequency is $E=mp$. That raises a question: when actual observed counts differ from expected, how do you decide if the gap is "within normal variability" or genuinely suspicious? This card answers it → gaps up to about $1$–$2$ standard deviations from the expected value are typical; gaps beyond $2$ standard deviations warrant further investigation.

The expected frequency tells you what to predict; the observed frequency is what actually happens. The gap between them is natural variability. A useful rule of thumb: differences of up to about $1$–$2$ standard deviations from the expected value are common, while gaps larger than $3$ SDs are unusual enough to investigate.

Example: A coin is tossed $400$ times. Expected number of heads $= 400 \times 0.5 = 200$.

• Variance: $npq = 400 \times 0.5 \times 0.5 = 100$, so SD $= 10$.
• Observing $195$ heads (deviation $5$, that is $0.5$ SDs) — completely typical.
• Observing $230$ heads (deviation $30$, that is $3$ SDs) — unusual; the coin may not be fair.

So expected frequencies are not just calculations — they give you a baseline to compare real-world data against.

The expected frequency tells you what to predict; the observed frequency is what actually happens. The gap between them is natural variability. A useful rule of thumb: differences of up to about $1$–$2$ standard deviations from the expected value are...

Pause — copy the rule of thumb: observed frequency within $\pm 2$ SD of expected is normal variation; a larger gap suggests the model may not hold into your book.