Probability Revision Notes

Probability measures how likely an event is to happen, expressed as a number between 0 and 1 inclusive. The symbol P(A) denotes the probability that event A occurs.

For example, if you toss a fair six-sided die, the probability of rolling a 4 is P(rolling a 4) = 1/6 because there is one favourable outcome (rolling a 4) out of six equally likely outcomes.

This notation helps us clearly state and work with probabilities in calculations and explanations.

• P(A) is always between 0 and 1.
• P(A) = 0 means event A cannot happen.
• P(A) = 1 means event A is certain to happen.

The complement of event A, written as A′ (read as 'not A'), represents all outcomes where event A does not occur.

The probability of the complement is denoted by P(A′). Since either event A happens or it does not, the probabilities must add up to 1.

This relationship is expressed as: P(A′) = 1 − P(A). For example, if the probability of raining today (event A) is 0.3, then the probability it does not rain (A′) is 1 − 0.3 = 0.7.

• P(A) + P(A′) = 1
• P(A′) is useful when it is easier to calculate the probability of an event not happening.

The symbol P(A ∩ B) represents the probability that both events A and B happen at the same time. This is called the intersection of events A and B.

For example, consider drawing one card from a standard deck. Let event A be 'the card is a heart' and event B be 'the card is a queen'. The only card that satisfies both is the queen of hearts.

Since there is 1 queen of hearts in 52 cards, P(A ∩ B) = 1/52 ≈ 0.0192.

• P(A ∩ B) means 'A and B' both occur.
• Calculating P(A ∩ B) depends on whether events are independent or mutually exclusive (covered in other objectives).

The symbol P(A ∪ B) denotes the probability that event A or event B or both occur. This is called the union of events A and B.

For example, using the same deck of cards, let event A be 'the card is a heart' and event B be 'the card is a queen'.

There are 13 hearts and 4 queens, but the queen of hearts is counted twice if we add directly. So, P(A ∪ B) = P(A) + P(B) − P(A ∩ B) = 13/52 + 4/52 − 1/52 = 16/52 ≈ 0.3077.

• P(A ∪ B) means 'A or B or both' occur.
• Use the formula P(A ∪ B) = P(A) + P(B) − P(A ∩ B) to avoid double counting.

These symbols provide a concise language to describe and calculate probabilities in various contexts.

Understanding and correctly applying P(A), P(A′), P(A ∩ B), and P(A ∪ B) is essential for solving probability problems accurately.

Always interpret the symbols carefully and check that probabilities are between 0 and 1, and that complementary probabilities sum to 1.

• Use P(A) to state the chance of event A.
• Use P(A′) to find the chance event A does not happen.
• Use P(A ∩ B) to find the chance both A and B happen.
• Use P(A ∪ B) to find the chance either A or B or both happen.

Probability quantifies how likely an event is to happen. It is a number between 0 and 1, where 0 means the event cannot happen and 1 means it is certain to happen. Probabilities can be expressed as fractions (e.g., 1/4), decimals (e.g., 0.25), or percentages (e.g., 25%).

To calculate the probability of an event, you divide the number of favourable outcomes by the total number of possible outcomes, assuming all outcomes are equally likely.

• Probability ranges from 0 (impossible) to 1 (certain).
• Express as fraction, decimal, or percentage.
• Probability formula: P(event) = (number of favourable outcomes) / (total number of outcomes).

When selecting items from a set, the total number of possible outcomes depends on whether the selection is made with or without replacement.

With replacement: After selecting an item, it is put back before the next selection. This means the total number of possible outcomes stays the same for each selection.

Without replacement: After selecting an item, it is not returned. This reduces the total number of possible outcomes for the next selection.

• With replacement: total outcomes remain constant.
• Without replacement: total outcomes decrease after each selection.

Suppose you have a bag containing 5 red balls and 3 blue balls (total 8 balls). You draw one ball, note its colour, then put it back before drawing again.

What is the probability of drawing a red ball on the first draw?

Using the formula: P(red) = number of red balls / total balls = 5/8 = 0.625 = 62.5%. Since the ball is replaced, the probability remains the same for the second draw.

• Calculate probability by dividing favourable outcomes by total outcomes.
• Replacement keeps total outcomes constant for each draw.

Using the same bag with 5 red and 3 blue balls, you draw one ball and do not replace it before drawing a second ball.

What is the probability of drawing a red ball on the first draw? It is 5/8 as before.

If the first ball drawn is red and not replaced, what is the probability that the second ball drawn is also red? Now there are 4 red balls left and only 7 balls total, so P(second red | first red) = 4/7 ≈ 0.571 or 57.1%.

• Without replacement, total outcomes decrease after each draw.
• Probabilities change depending on previous outcomes.

Probabilities can be expressed in three common ways: fractions, decimals, and percentages.

Fraction: Shows the ratio of favourable outcomes to total outcomes, e.g., 3/10.

Decimal: The fraction converted to decimal form, e.g., 0.3 for 3/10, often rounded to two or three decimal places as appropriate for clarity and precision in answers, especially in exams or reports. Use exact decimals when possible to avoid rounding errors in further calculations. For example, 1/4 = 0.25 exactly, but 1/3 ≈ 0.333 (rounded). Percentage: The decimal multiplied by 100 and followed by the % symbol, e.g., 30%. This is useful for communicating probabilities in everyday contexts or when percentages are requested in exam questions.

• Fraction: exact ratio of favourable to total outcomes.
• Decimal: fraction converted to decimal form, rounded appropriately.
• Percentage: decimal × 100 with % sign.

Calculating probability in simple scenarios involves counting the number of favourable outcomes and dividing by the total number of possible outcomes.

It is essential to identify whether selections are made with replacement (total outcomes stay the same) or without replacement (total outcomes decrease).

Express probabilities clearly as fractions, decimals, or percentages depending on the context or exam requirements.

• Probability = favourable outcomes ÷ total outcomes.
• With replacement: total outcomes constant.
• Without replacement: total outcomes decrease.
• Express answers as fraction, decimal, or percentage as required.

In probability, every event has a complementary event, which represents all outcomes where the original event does not happen. For example, if event A is 'rolling a 4 on a six-sided die,' then the complementary event A′ is 'not rolling a 4,' which includes rolling 1, 2, 3, 5, or 6.

The total probability of all possible outcomes in any experiment is always 1. Since event A and its complement A′ cover all possible outcomes without overlap, their probabilities add up to 1. This relationship is the foundation for calculating complementary probabilities.

• Complementary events are mutually exclusive and exhaustive.
• P(A) + P(A′) = 1, where P(A) is the probability of event A occurring.
• P(A′) represents the probability of event A not occurring.

The formula to find the probability of an event not occurring is:

P(A′) = 1 - P(A) where: P(A) is the probability of the event occurring, and P(A′) is the probability of the event not occurring.

This formula is useful when it is easier to calculate the probability of the event happening than not happening. By subtracting P(A) from 1, you find the probability of the complementary event.

• Formula: P(A′) = 1 - P(A)
• Used when P(A) is known or easier to find.
• Ensures probabilities sum to 1.

Suppose the probability that it will rain tomorrow is 0.3. To find the probability that it will not rain, use the complementary probability formula:

P(not rain) = 1 - P(rain) = 1 - 0.3 = 0.7. Therefore, there is a 70% chance it will not rain tomorrow.

This example shows how complementary probability provides a quick way to find the likelihood of the opposite event.

• Given P(rain) = 0.3
• Calculate P(not rain) = 1 - 0.3 = 0.7
• Interpretation: 70% chance of no rain

Complementary probability is widely applicable in various scenarios, such as games, quality control, and risk assessment. For instance, if the probability of drawing a red card from a deck is 0.5, then the probability of not drawing a red card is 1 - 0.5 = 0.5.

It is important to ensure that the event and its complement cover all possible outcomes without overlap and that probabilities are between 0 and 1.

• Useful in games, experiments, and decision-making.
• Always check that P(A) is valid (0 ≤ P(A) ≤ 1).
• Complement covers all outcomes not in event A.

A frequent error is forgetting that probabilities must sum to 1, leading to incorrect complementary probabilities. Another mistake is confusing the event with its complement, especially when interpreting word problems.

Always verify that the event and its complement are mutually exclusive and collectively exhaustive. Use clear notation to distinguish P(A) from P(A′).

• Do not add probabilities of overlapping events when using the complement formula.
• Check that P(A) is correctly calculated before subtracting from 1.
• Use clear event definitions to avoid confusion.

The addition rule is a fundamental principle in probability that helps us find the chance that at least one of two events happens. When we want to know the probability that event A or event B occurs, we write this as P(A ∪ B), which means the probability of the union of A and B.

If events A and B can happen at the same time, simply adding their probabilities, P(A) + P(B), counts the overlap twice. To correct this, we subtract the probability that both events happen together, P(A ∩ B). This gives the formula:

P(A ∪ B) = P(A) + P(B) - P(A ∩ B). This ensures the combined probability is accurate.

• P(A ∪ B): Probability that A or B or both occur.
• P(A): Probability that event A occurs.
• P(B): Probability that event B occurs.
• P(A ∩ B): Probability that both A and B occur simultaneously.

When events A and B overlap, the outcomes where both happen are included in both P(A) and P(B). Adding these probabilities counts these overlapping outcomes twice. Subtracting P(A ∩ B) removes this double counting, giving the correct total probability.

For example, if you add the probabilities of drawing a red card or a king from a deck of cards, the red kings are counted twice unless you subtract the probability of drawing a red king.

• Overlap causes double counting if not corrected.
• Subtracting P(A ∩ B) adjusts for this overlap.
• This is essential when events are not mutually exclusive.

Suppose you roll a six-sided die. Let event A be 'rolling an even number' and event B be 'rolling a number greater than 3'.

P(A): Even numbers are 2, 4, 6, so P(A) = 3/6 = 0.5.

P(B): Numbers greater than 3 are 4, 5, 6, so P(B) = 3/6 = 0.5, too.

• Find P(A ∩ B): Numbers that are even and greater than 3 are 4 and 6, so P(A ∩ B) = 2/6 ≈ 0.333.
• Use the addition rule: P(A ∪ B) = P(A) + P(B) - P(A ∩ B) = 0.5 + 0.5 - 0.333 = 0.667.
• Interpretation: There is a 66.7% chance of rolling an even number or a number greater than 3.

If events A and B cannot happen at the same time, they are called mutually exclusive. In this case, P(A ∩ B) = 0 because there is no overlap.

The addition rule simplifies to: P(A ∪ B) = P(A) + P(B).

For example, when drawing a single card, event A = 'drawing a heart' and event B = 'drawing a club' are mutually exclusive because a card cannot be both suits.

• No overlap means no subtraction needed.
• Simplifies calculations for mutually exclusive events.
• Important to identify this to avoid errors.

To apply the addition rule correctly: first identify the events and their probabilities.

Next, determine if the events can occur together (find P(A ∩ B)).

Then substitute values into the formula and calculate carefully, ensuring fractions, decimals or percentages are consistent in form and units are clear if relevant (e.g., time or trials).

• Check if events overlap before applying the formula.
• Calculate or find P(A), P(B), and P(A ∩ B) accurately.
• Use consistent probability formats (fraction, decimal, percentage).
• Interpret the final probability in context.

Mutually exclusive events are two or more events that cannot happen at the same time. If one event occurs, the other cannot. This means their outcomes do not overlap.

For example, when rolling a single six-sided die, the event 'rolling a 2' and the event 'rolling a 5' are mutually exclusive because the die can only show one number at a time.

Because these events cannot happen together, the probability that either event A or event B occurs is simply the sum of their individual probabilities. This is expressed as: P(A ∪ B) = P(A) + P(B).

• Mutually exclusive means no shared outcomes.
• Events cannot occur simultaneously.
• Use addition rule for probability of either event.

To calculate the probability of either event A or event B occurring when they are mutually exclusive, add their probabilities.

For example, consider drawing a single card from a standard deck of 52 cards. Let event A be drawing a heart, and event B be drawing a club. Since a card cannot be both a heart and a club, these events are mutually exclusive.

P(A) = number of hearts / total cards = 13/52 = 1/4. P(B) = number of clubs / total cards = 13/52 = 1/4. Therefore, P(A ∪ B) = P(A) + P(B) = 1/4 + 1/4 = 1/2. So, there is a 50% chance of drawing either a heart or a club.

• Add probabilities for mutually exclusive events.
• Example: P(heart or club) = 1/4 + 1/4 = 1/2.

Independent events are events where the occurrence of one does not affect the probability of the other occurring.

For example, flipping a coin and rolling a die are independent events because the result of the coin flip does not influence the die roll.

When events are independent, the probability that both events A and B occur together is found by multiplying their individual probabilities: P(A ∩ B) = P(A) × P(B).

• Independent means no influence between events.
• Probability of both events occurring is product of their probabilities.
• Use multiplication rule for independent events.

To find the probability that both independent events occur, multiply their probabilities.

For example, if you flip a coin and roll a die, what is the probability of getting heads on the coin and a 4 on the die?

P(heads) = 1/2, P(4 on die) = 1/6. Since these are independent, P(heads and 4) = 1/2 × 1/6 = 1/12. So, there is an 8.33% chance of both events happening together.

• Multiply probabilities for independent events.
• Example: P(heads and 4) = 1/2 × 1/6 = 1/12.

It is important to understand that mutually exclusive and independent events are different concepts.

Mutually exclusive events cannot happen at the same time, so their joint probability P(A ∩ B) = 0.

Independent events can happen together, and the occurrence of one does not change the probability of the other, so P(A ∩ B) = P(A) × P(B). If two events are mutually exclusive and one has a non-zero probability, they cannot be independent because P(A ∩ B) would be zero, not the product of their probabilities.

• Mutually exclusive: events cannot co-occur; P(A ∩ B) = 0.
• Independent: events can co-occur; P(A ∩ B) = P(A) × P(B).
• Events cannot be both mutually exclusive and independent unless one has zero probability.

Conditional probability measures the chance that an event A happens when it is known that another event B has already occurred. Instead of considering all possible outcomes, we focus only on those outcomes where event B occurs, then find the proportion of these where event A also happens.

This approach helps us understand how the occurrence of one event affects the likelihood of another, which is different from simply finding the chance of A or B happening independently.

• Conditional probability is the probability of A given B has occurred.
• Only outcomes where B happens are considered when calculating this probability.
• It shows how the knowledge of B affects the chance of A.

To calculate the conditional probability of A given B, we first identify all outcomes where B occurs. Then, among these, we count how many outcomes also satisfy A. The conditional probability is the fraction of B's outcomes that also include A.

Although the formula P(A | B) = P(A ∩ B) / P(B) is a concise way to express this, you do not need to memorise it. Instead, focus on the reasoning: the probability of A given B is the number of outcomes where both A and B happen divided by the number of outcomes where B happens.

This method applies to any basic situation where probabilities are known or can be found by counting outcomes.

• Identify all outcomes where event B occurs.
• Count how many of these outcomes also satisfy event A.
• Divide the number of outcomes with both A and B by the number of outcomes with B.
• Express the result as a fraction, decimal, or percentage as appropriate.

Suppose you have a standard deck of 52 playing cards. Event B is drawing a red card (hearts or diamonds), and event A is drawing a queen. We want to find the probability of drawing a queen given that the card drawn is red.

First, find the number of red cards: there are 26 red cards in total.

Next, find how many red cards are queens: there are 2 red queens (queen of hearts and queen of diamonds). Since we know the card is red, the sample space reduces to these 26 cards. The conditional probability is therefore the number of red queens divided by the number of red cards: 2/26 = 1/13 ≈ 0.0769 or 7.69%.

• Event B: card is red (26 cards).
• Event A: card is a queen (4 cards total, 2 red).
• Conditional probability P(A | B) = number of red queens / number of red cards = 2/26 = 1/13.
• Interpretation: If you know the card is red, there is about a 7.7% chance it is a queen.

Conditional probability results must be interpreted carefully in context. The probability of A given B can be very different from the probability of A alone.

For example, the probability of drawing a queen from the full deck is 4/52 = 1/13, which coincidentally equals the conditional probability in the previous example. However, this is not always the case; often, knowing that B has occurred changes the likelihood of A.

Understanding this helps in real-life situations, such as medical testing or risk assessment, where the chance of an event depends on prior conditions or information.

• Conditional probability depends on the condition that event B has occurred.
• It can be higher or lower than the unconditional probability of A.
• Always interpret the result in the context of the given condition.

Conditional probability differs from independent events, where the occurrence of one event does not affect the probability of another. If A and B are independent, then P(A | B) = P(A).

It also differs from mutually exclusive events, where two events cannot happen at the same time, so P(A ∩ B) = 0 and thus P(A | B) = 0.

Recognising these differences is important to avoid mistakes when calculating or interpreting probabilities.

• If events are independent, knowing B does not change the probability of A.
• If events are mutually exclusive, the conditional probability of A given B is zero.
• Conditional probability measures dependence between events.

A tree diagram is a graphical tool used to display all possible outcomes of a sequence of events in an organised way. It starts from a single point and branches out for each possible outcome of the first event. From each of these branches, further branches represent the possible outcomes of the next event, and so on. This structure helps to visualise combined events and their probabilities clearly, especially when events occur in sequence.

• Start with one point representing the start of the experiment.
• Branches represent possible outcomes of each event.
• Each level corresponds to one event in the sequence.

Each branch in a tree diagram is labelled with the probability of that particular outcome occurring, given the path taken so far. For the first event, these probabilities are unconditional. For subsequent events, probabilities may depend on previous outcomes (dependent events) or remain the same (independent events). The sum of the probabilities of all branches from a single point must always be 1, reflecting that one of the possible outcomes must occur.

• Label each branch with the probability of that outcome.
• Probabilities from one event’s branches sum to 1.
• Use conditional probabilities if events are dependent.

To find the probability of a combined event (a sequence of outcomes), multiply the probabilities along the branches leading to that event. This multiplication rule follows from the definition of combined probabilities for sequential events. The product gives the probability of that exact sequence occurring. Summing the probabilities of branches leading to different sequences can find probabilities of events defined by multiple outcomes.

• Multiply probabilities along the branches to find combined event probability.
• For example, P(A then B) = P(A) × P(B|A) if dependent.
• Sum probabilities of relevant branches to find probabilities of combined events with multiple outcomes.

Consider tossing two coins in sequence and finding the probability of getting one head and one tail in any order. The first event is the first coin toss with outcomes Head (H) or Tail (T), each with probability 1/2. The second event is the second coin toss, also H or T with probability 1/2, independent of the first.

• Draw branches from the start: H (1/2) and T (1/2).
• From each branch, draw two branches: H (1/2) and T (1/2).
• Calculate combined probabilities by multiplying along branches, e.g., P(H then T) = 1/2 × 1/2 = 1/4.
• The combined event 'one head and one tail' includes sequences H then T and T then H, so sum: 1/4 + 1/4 = 1/2.

When the outcome of one event affects the probabilities of the next, the events are dependent. Tree diagrams can still be used by adjusting the probabilities on branches accordingly. For example, if drawing cards without replacement, the probability of the second card depends on the first card drawn.

• Adjust probabilities on branches to reflect dependence.
• Multiply along branches as usual to find combined probabilities.
• Example: Drawing two cards without replacement from a deck.