Relative & Expected Frequency

Relative Frequency

Relative frequency is a way to express how often an event happens compared to the total number of trials or observations. It is calculated as the ratio of the frequency of an event to the total number of trials.

For example, if you toss a coin 10 times and get 6 heads, the relative frequency of heads is $\frac{6}{10} = 0.6$.

The formula for relative frequency is:

$$\text{Relative frequency} = \frac{\text{Frequency of event}}{\text{Total number of trials}}$$

Relative frequency can be used to estimate the probability of an event occurring, especially when the theoretical probability is unknown or difficult to calculate.

For instance, if you toss a coin 50 times and it lands on heads 28 times, the relative frequency of getting heads is:

$$\frac{28}{50} = 0.56$$

This means the estimated probability of getting heads from this experiment is 0.56, or $56\%$.

Expected Frequency

Expected frequency is the number of times an event is predicted to occur based on its probability and the total number of trials.

It is calculated by multiplying the probability of the event by the total number of trials:

$$\text{Expected frequency} = \text{Probability of event} \times \text{Total number of trials}$$

Expected frequency helps to predict how often an event should happen if the experiment is repeated many times under the same conditions.

For example, if the probability of rolling a 3 on a fair six-sided die is $\frac{1}{6}$, and you roll the die 60 times, the expected frequency of rolling a 3 is:

$$\frac{1}{6} \times 60 = 10$$

This means you would expect to roll a 3 about 10 times in 60 rolls.

Comparing expected frequency with observed frequency (the actual number of times the event occurs) can show how close an experiment is to the theoretical prediction.

Applications in Probability

Relative frequency is commonly used to estimate probabilities from experimental data. When an experiment is repeated many times, the relative frequency of an event tends to get closer to the theoretical probability.

Theoretical probability is the probability calculated based on known possible outcomes, assuming all outcomes are equally likely.

Experimental probability (or relative frequency) may differ from theoretical probability due to chance or small sample sizes.

As the number of trials increases, the relative frequency usually becomes a better estimate of the theoretical probability. This is known as the Law of Large Numbers.

Expected frequency is useful for predicting how many times an event should occur in a given number of trials, helping to check if experimental results are reasonable.

For example, if you flip a fair coin 100 times, the theoretical probability of heads is $\frac{1}{2}$. The expected frequency of heads is:

$$\frac{1}{2} \times 100 = 50$$

If you actually get 55 heads, the relative frequency is $\frac{55}{100} = 0.55$, which is close to the theoretical probability of 0.5.