Relative frequency, the effect of more trials and expected frequency np (Year 12 Relative frequency and probability, 2024 Mathematics Standard 1 syllabus)

Relative frequency is the number of times the event happened divided by the number of trials.

Marker's note: one mark for the correct fraction, one for the decimal or percentage. The denominator is the number of trials ($150$), not the number of times it landed point down.

Use expected frequency $np$, with $n = 120$ and $p = 0.15$.

Marker's note: one mark for the substitution, one for $18$. The answer is an expected value, not a guarantee.

(a) Expected number

$p = \frac{1}{6}$, so $np = 300 \times \frac{1}{6} = 50$.

(b) Relative frequency

$\frac{38}{300} = \frac{19}{150} \approx 0.127$.

Compare

The theoretical probability is $\frac{1}{6} \approx 0.167$, so the experimental result is lower than expected. With only $300$ rolls, a difference of $12$ sixes from $50$ is quite possible by chance.

Marker's note: one mark for $50$, one for $0.127$, one for a sensible comparison. A comment that a short run can differ from the theory without the die being unfair is rewarded.

Relative frequencies.

What it shows. As the number of trials increases, the relative frequency gets closer to the theoretical probability of $0.5$. Results from a larger number of trials are a more reliable estimate of probability.

Marker's note: one mark for the three values, one for the trend, one for stating that more trials give a better estimate.

Faulty. $p = 0.025$, so $np = 800 \times 0.025 = 20$.

Not faulty. $P(\text{not faulty}) = 1 - 0.025 = 0.975$, so $800 \times 0.975 = 780$. Check: $20 + 780 = 800$.

Marker's note: one mark for converting $2.5\%$ to $0.025$, one for $20$, one for $780$.

(a) Relative frequencies

Red $\frac{148}{400} = 0.37$, blue $\frac{112}{400} = 0.28$, green $\frac{140}{400} = 0.35$. These add to $1$.

(b) Equally likely

Each colour has $p = \frac{1}{3}$, so the expected frequency is $400 \times \frac{1}{3} = 133.\overline{3}$, about $133$ spins.

(c) Estimate for $1000$ spins

Using $0.37$ as the probability, $1000 \times 0.37 = 370$ reds.

Marker's note: one mark for each of (a) and (b), and two for (c), one for using the relative frequency as $p$ and one for $370$. Blue is noticeably below $133$, so it is reasonable to suspect the spinner is not fair, but $400$ spins is not enough to be certain.

Relative frequency from the sample

$\frac{36}{90} = \frac{2}{5} = 0.4$.

Expected number in the town

$4500 \times 0.4 = 1800$ residents.

Why it may be inaccurate

The sample of $90$ is small compared with $4500$, so another sample could give a different proportion. A larger random sample gives a better estimate.

Marker's note: one mark for $0.4$, one for $1800$, one for a valid reason such as a small sample or a possibly unrepresentative sample.