IB Mathematics AA Concepts of trial, outcome Study Notes
IB Mathematics AA Concepts of trial outcome data Study Notes
IB Mathematics AA Concepts of trial outcome Notes Offer a clear explanation of Concepts of trial outcome, including various formula, rules, exam style questions as example to explain the topics. Worked Out examples and common problem types provided here will be sufficient to cover for topic Concepts of trial outcome.
Concepts of Trial, Outcome, Relative Frequency, Sample Space, and Event
Trial
A trial is a single attempt or performance of a random experiment. It is the action we do to observe an outcome.
Examples:
- Tossing a coin once is a trial.
- Rolling a die once is a trial.
- Selecting one card from a deck is a trial.
Outcome
An outcome is the result we get from a trial.
Examples:
- The outcome of tossing a coin could be head or tail.
- The outcome of rolling a die could be any number from 1 to 6.
- The outcome of drawing a card could be the 5 of hearts or any other specific card.
Equally Likely Outcomes
Outcomes are called equally likely if each has the same chance of occurring. This happens when there is no bias or unfairness in the trial.
Examples:
- In tossing a fair coin, head and tail are equally likely; probability of head = probability of tail = 0.5.
- In rolling a fair six-sided die, each face (1 to 6) is equally likely; probability = \( \frac{1}{6} \) for each.
Non-example: If a die is weighted to favour 6, then the outcomes are not equally likely.
Relative Frequency
The relative frequency of an event is the ratio:
\( \frac{\text{Number of times an event occurs}}{\text{Total number of trials}} \)
Relative frequency is used to estimate probability from experimental data, especially when the theoretical probability is unknown.
Examples:
- A die is rolled 200 times and the number 4 appears 35 times.
Relative frequency of 4 = \( \frac{35}{200} = 0.175 \). - A coin is tossed 100 times and lands heads 47 times.
Relative frequency of heads = \( \frac{47}{100} = 0.47 \).
Note: As the number of trials increases, relative frequency approaches the theoretical probability (Law of Large Numbers).
Sample Space (U)
The sample space (denoted by \( U \) or \( S \)) is the set of all possible outcomes of a trial.
Examples:
- Tossing a coin: \( U = \{ \text{Head}, \text{Tail} \} \)
- Rolling a die: \( U = \{ 1, 2, 3, 4, 5, 6 \} \)
- Drawing a card: \( U \) = {all 52 cards in the deck}
The sample space helps define events and calculate probabilities.
Event
An event is a set of one or more outcomes that we are interested in. Events are subsets of the sample space.
Examples:
- Event A: Rolling an even number on a die.
\( A = \{ 2, 4, 6 \} \) - Event B: Tossing a head.
\( B = \{ \text{Head} \} \) - Event C: Drawing a red card from a deck.
\( C = \{ \text{All hearts and diamonds} \} \)
Events can be:
- Simple event: contains a single outcome (e.g. rolling a 5).
- Compound event: contains multiple outcomes (e.g. rolling an even number).
Example:
A fair coin is tossed 3 times. List the sample space. If this trial is repeated 100 times and the event “exactly 2 heads” occurs 36 times, calculate the relative frequency of getting exactly 2 heads.
▶️ Answer/Explanation
The sample space \( U \) for tossing 3 coins is:
U = { HHH, HHT, HTH, THH, HTT, THT, TTH, TTT }
There are 8 possible outcomes.
outcomes for exactly 2 heads
The event E = “exactly 2 heads” includes:
E = { HHT, HTH, THH }
Relative frequency = \( \frac{\text{Number of times event occurred}}{\text{Total number of trials}} \)
Relative frequency = \( \frac{36}{100} = 0.36 \)
Probability of an Event
Probability of an Event
The probability of an event A happening is given by the formula:
\( P(A) = \frac{n(A)}{n(U)} \)
where:
- \( P(A) \) is the probability of event A.
- \( n(A) \) is the number of outcomes that satisfy event A.
- \( n(U) \) is the total number of possible outcomes in the sample space U.
Conditions:
- The outcomes must be equally likely.
- The outcomes must be from a well-defined sample space.
Example:
A fair 6-sided die is rolled. Find the probability of getting an even number.
▶️ Answer/Explanation
Step 1: Identify the sample space
The sample space is:
\( U = \{ 1, 2, 3, 4, 5, 6 \} \)
Number of outcomes: \( n(U) = 6 \)
Step 2: Identify event A
Event A = getting an even number:
\( A = \{ 2, 4, 6 \} \)
Number of favourable outcomes: \( n(A) = 3 \)
Step 3: Calculate probability
Using the formula:
\( P(A) = \frac{n(A)}{n(U)} = \frac{3}{6} = 0.5 \)
Conclusion: The probability of getting an even number is 0.5.
Complementary Events: A and A′ (Not A)
Complementary Events: A and A′ (Not A)
In probability, for any event \( A \), the complementary event (denoted by \( A’ \) or “not A”) is the event that consists of all outcomes in the sample space \( U \) that are not in \( A \).
Key property:
\( P(A) + P(A’) = 1 \)
or equivalently,
\( P(A’) = 1 – P(A) \)
This means the probability that event \( A \) does not occur is \( 1 – P(A) \).
Example:
A fair die is rolled. Find the probability of getting an even number and the probability of not getting an even number.
▶️ Answer/Explanation
\( U = \{ 1, 2, 3, 4, 5, 6 \}, \quad n(U) = 6 \)
\( A = \{ 2, 4, 6 \}, \quad n(A) = 3 \)
\( P(A) = \frac{n(A)}{n(U)} = \frac{3}{6} = 0.5 \)
\( P(A’) = 1 – P(A) = 1 – 0.5 = 0.5 \)
Conclusion: The probability of not getting an even number is 0.5.
Expected Number of Occurrences
Expected Number of Occurrences
The expected number of occurrences of an event A in a series of trials is a prediction of how many times the event is likely to happen based on its probability.
\( \text{Expected number of occurrences} = P(A) \times \text{number of trials} \)
Where:
- \( P(A) \) is the probability of event A occurring in a single trial.
- The number of trials is the total times the experiment is performed.
This is also known as the mean or expected frequency of the event.
Example:
A fair die is rolled 120 times. Find the expected number of times a 4 will occur.
▶️ Answer/Explanation
Event A = getting a 4.
\( P(A) = \frac{1}{6} \)
\( \text{Expected number} = P(A) \times \text{number of trials} \)
\( = \frac{1}{6} \times 120 \)
\( = 20 \)
Conclusion: We expect the number 4 to appear 20 times in 120 rolls of a fair die.
Experimental Probability
Experimental probability is the probability of an event based on actual results from performing trials or experiments.
\( P(A) = \frac{\text{Number of times event A occurs}}{\text{Total number of trials}} \)
The experimental probability may change as more trials are performed. It is also called empirical probability.
Theoretical Probability
Theoretical probability is the probability of an event based on reasoning, without conducting any trials. It assumes that all outcomes are equally likely.
\( P(A) = \frac{n(A)}{n(U)} \)
where:
- \( n(A) \) = number of favourable outcomes
- \( n(U) \) = total number of possible outcomes
Theoretical probability stays the same unless conditions change.
Example:
A coin is tossed 50 times. It lands heads 28 times. Compare the experimental probability of getting a head to the theoretical probability.
▶️ Answer/Explanation
Experimental probability
\( P(\text{Head}) = \frac{\text{Number of heads}}{\text{Number of trials}} \)
\( = \frac{28}{50} = 0.56 \)
Theoretical probability
For a fair coin:
\( P(\text{Head}) = \frac{1}{2} = 0.5 \)
Conclusion: The experimental probability (0.56) is close to the theoretical probability (0.5), but not exactly the same. More trials might bring it closer.