Relative and Expected Frequency (HSC SSCE Mathematics Standard): Revision Notes

Worked Example: Finding Relative Frequency

Let's look at an experiment where two coins are tossed repeatedly, and we record the number of heads that appear.

Question: Find the relative frequency of obtaining 0 heads, 1 head, and 2 heads.

Solution:

First, we need to find the total number of trials:

• Total $= 100 + 192 + 108 = 400$

For 0 heads:

• Apply the formula: $\text{Relative frequency} = \frac{f}{n}$
• Substitute values: $\text{Relative frequency} = \frac{100}{400}$
• Simplify: $\frac{100}{400} = \frac{1}{4} = 0.25 = 25\%$
• Answer: The relative frequency of getting 0 heads is $0.25$ or $25\%$

For 1 head:

• Apply the formula: $\text{Relative frequency} = \frac{f}{n}$
• Substitute values: $\text{Relative frequency} = \frac{192}{400}$
• Simplify: $\frac{192}{400} = \frac{12}{25} = 0.48 = 48\%$
• Answer: The relative frequency of getting 1 head is $0.48$ or $48\%$

For 2 heads:

• Apply the formula: $\text{Relative frequency} = \frac{f}{n}$
• Substitute values: $\text{Relative frequency} = \frac{108}{400}$
• Simplify: $\frac{108}{400} = \frac{27}{100} = 0.27 = 27\%$
• Answer: The relative frequency of getting 2 heads is $0.27$ or $27\%$

Worked Example: Performing a Simulation

Let's simulate the results of 50 cricket matches in a season, where there are three possible outcomes for each match.

Given information:

• Probability of winning a match: $30\%$
• Probability of drawing a match: $50\%$
• Probability of losing a match: $20\%$

Task: Use 10 playing cards (numbered 1 to 10) to simulate 50 matches and record the results.

Solution:

First, we assign cards to each outcome based on their probabilities:

• Win: $30\%$ of 10 cards $= 3$ cards → Assign cards 1, 2, and 3 to "Win"
• Draw: $50\%$ of 10 cards $= 5$ cards → Assign cards 4, 5, 6, 7, and 8 to "Draw"
• Lose: $20\%$ of 10 cards $= 2$ cards → Assign cards 9 and 10 to "Lose"

Then, we conduct the simulation:

• Select a card at random
• Record the result
• Replace the card
• Repeat 50 times

Here are the results in a frequency table with relative frequencies:

Worked Example: Finding Expected Frequency

Two coins are tossed 120 times and the results are recorded.

Question a: What is the expected frequency for two heads?

Solution:

• First, find the probability of getting two heads
• Possible outcomes when tossing two coins: HH, HT, TH, TT (4 outcomes)
• Favourable outcomes for two heads: HH (1 outcome)
• Probability: $P(\text{HH}) = \frac{1}{4}$
• Apply the expected frequency formula: $\text{Expected frequency} = np$
• Substitute: $\text{Expected frequency} = \frac{1}{4} \times 120$
• Calculate: $\frac{1}{4} \times 120 = 30$
• Answer: The expected frequency for two heads is 30

Question b: What is the expected frequency for a head and a tail?

Solution:

• Find the probability of getting one head and one tail
• Favourable outcomes: HT and TH (2 outcomes)
• Total outcomes: 4
• Probability: $P(\text{HT or TH}) = \frac{2}{4} = \frac{1}{2}$
• Apply the formula: $\text{Expected frequency} = np$
• Substitute: $\text{Expected frequency} = \frac{1}{2} \times 120$
• Calculate: $\frac{1}{2} \times 120 = 60$
• Answer: The expected frequency for one head and one tail is 60