Understanding expected frequencies is crucial in predicting the outcomes of various events based on their probabilities. This concept allows us to estimate how often an event will occur over a number of trials, providing a foundational tool in the study of probability and statistics.

Introduction to Expected Frequencies

Expected frequency is a statistical measure used to predict how often an event will occur over a certain number of trials. It is calculated using the formula:

This concept is fundamental in probability theory and helps in making informed predictions about the outcomes of different scenarios.

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Calculating Expected Frequencies

Formula:

Where:

• $P(E)$ is the probability of the event occurring
• $N$ is the total number of trials

Example 1: Coin Toss

Given: A fair coin is tossed 100 times.

Find: Expected frequency of getting heads.

• $P(\text{Heads}) = 0.5$
• $N = 100$

Therefore, you would expect heads to appear 50 times out of 100 tosses.

Example 2: Dice Rolls

Given: A fair six-sided die is rolled 60 times.Find: Expected frequency of rolling a 4.

• $P(4) = \dfrac{1}{6}$
• $N = 60$

Thus, in 60 rolls of a die, a 4 is expected to appear 10 times.

Applying Expected Frequencies

Expected frequencies can predict outcomes in various scenarios, from simple games to complex scientific experiments.

Example 1: Drawing Balls from a Bag

Given: A bag contains 5 red, 3 blue, and 2 green balls. Balls are drawn 100 times with replacement.

Find: Expected frequency of drawing a red ball.

• $P(\text{Red}) = \frac{5}{10} = 0.5$
• $N = 100$

Expect to draw a red ball 50 times out of 100 draws.

Example 2: School Survey

Given: 70% of students prefer online classes.

Find: Expected number of students preferring online classes out of 200 surveyed.

• $P(\text{Online}) = 0.7$
• $N = 200$

Expect 140 out of 200 students to prefer online classes.

Practice Questions

Question 1

A spinner is divided into 5 equal sections, marked 1 through 5. If spun 500 times, what is the expected frequency of landing on section 3?

Solution:

Given a spinner divided into 5 equal sections and spun 500 times, we find the expected frequency of landing on section 3 as follows:

Therefore, the spinner is expected to land on section 3 100 times out of 500 spins.

Question 2

A school has 60% boys and 40% girls. In a random sample of 100 students, how many are expected to be boys?

Solution:

Given that a school has 60% boys and a random sample of 100 students is taken, the expected number of boys is:

Hence, in a sample of 100 students, 60 are expected to be boys.