Expected Number of Successes
When an experiment is repeated many times and outcomes are equally likely, probability can be used to predict the expected number of successes. This does not give an exact result but describes what is likely to happen in the long run.
What Is a Success
A success is the outcome or event you are interested in.
For example:
• getting a head when tossing a coin
• rolling a 6 on a die
• picking a red counter from a bag
Each repetition of the experiment is called a trial.
Understanding Expected Number
The expected number of successes is the number of times an event is predicted to occur after many trials.
It is based on:
• the probability of success in one trial
• the number of trials
The expected number is not guaranteed to happen exactly, but it is the average result over many repeats.
Expected does not mean certain
Calculating the Expected Number of Successes
The expected number of successes is found using:
$$
expected\ number\ =\ probability\ of\ success\ ×\ number\ of\ trials
$$
This formula only applies when:
• the probability stays the same each time
• outcomes are equally likely
• trials are independent
Using Probability Correctly
The probability must be written as a number between zero and one.
For example:
• probability of success is \( \frac{1}{2} \)
• number of trials is \( 20 \)
The expected number of successes is:
$$
\frac12×20
$$
This gives the long term average number of successes.
Interpreting the Result
The expected number:
• may not be a whole number
• represents an average over many experiments
For example, an expected number of \( 2.5 \) successes does not mean half a success occurs. It means that over many repeats, the average result is 2.5 successes per set of trials.
Actual results may be higher or lower in any one experiment.
Link to Long Term Behaviour
As the number of trials increases:
• results tend to get closer to the expected value
• relative frequency becomes more stable
This links expected number to experimental probability and long term stability.
Common Errors to Avoid
Common mistakes include:
• thinking the expected number must happen exactly
• using percentages without converting
• applying the method when outcomes are not equally likely
Always check the conditions before using the formula.
Key Points to Remember
A success is the outcome of interest.
Expected number predicts long term average results.
It is found by multiplying probability by number of trials.
The result may not be a whole number.
Actual results can vary from the expected number.
Understanding and using expected number of successes helps predict outcomes sensibly when experiments are repeated many times under fair and equally likely conditions.