Probability III. Vectors and Mechanics

Correlation measures how two variables move together. When one increases and the other also increases, they're positively correlated. When one increases while the other decreases, they're negatively correlated. No relationship means zero correlation.

Think about WAEC exam scores and study hours. Students who study more hours typically score higher marks—this is positive correlation. The strength ranges from -1 (perfect negative) through 0 (no relationship) to +1 (perfect positive).

In Nigeria, consider how rainfall and crop yield correlate positively in farming regions. More rain usually means better harvests. However, correlation doesn't mean one causes the other. Ice cream sales and accident rates both increase in summer, but ice cream doesn't cause accidents.

The correlation coefficient (r) quantifies this relationship mathematically. You'll calculate it using given data and interpret what the value means for real situations.

Probability simply means the chance that something will happen. It measures how likely an event is to occur, ranging from zero (impossible) to one (certain). Think of it as a way mathematicians measure uncertainty in everyday situations.

When you buy a raffle ticket at your local market in Lagos, you're dealing with probability. If 500 tickets are sold and only one wins the grand prize, your probability of winning is 1 out of 500, or 1/500. This fraction shows how likely success is. Probability helps us make informed decisions by showing us the realistic chances of different outcomes.

The basic formula is: Probability = (Number of favorable outcomes) ÷ (Total number of possible outcomes). Understanding this foundation is crucial because WAEC loves testing whether you can identify what counts as favorable outcomes versus total possibilities.

Relative frequency is simply how often something happens compared to the total number of times you tried it. When you flip a coin 100 times and get heads 45 times, the relative frequency of heads is 45÷100 = 0.45 or 45%. It's the actual result from your experiment, not the theoretical probability.

Think of it like a hawker selling pure water in Lagos. If she sells to 120 customers in a week and 84 buy her water, the relative frequency of making a sale is 84÷120 = 0.7 or 70%. This is what actually happened, not what should happen in theory.

As you repeat an experiment more times, your relative frequency usually gets closer to the theoretical probability. This is called the law of large numbers. With just 10 trials, relative frequency might jump around, but with 1000 trials, it stabilizes.

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Probability measures how likely something is to happen. When you have a simple sample space, you're dealing with all possible outcomes of an event that are equally likely to occur. To calculate probability, divide the number of favourable outcomes by the total number of possible outcomes.

Think of it this way: imagine a raffle draw at your school where 500 students buy tickets and 20 win prizes. Your probability of winning is 20 out of 500, which simplifies to 1 out of 25. This is straightforward because every ticket has equal chance of winning.

The formula is always: Probability = (Number of favourable outcomes) ÷ (Total number of possible outcomes). When tossing a fair coin, you have two equally likely outcomes—heads or tails—so the probability of getting heads is 1/2.

When two events can happen together or separately, you need to know when to add or multiply their probabilities. The addition rule applies when events are mutually exclusive—meaning they cannot occur at the same time. For example, when drawing one card from a deck, you cannot get both a heart and a spade simultaneously. You add the probabilities: P(heart or spade) = P(heart) + P(spade).

The multiplication rule applies when events are independent—one event's outcome doesn't affect the other. If you toss a coin twice, getting heads on the first toss doesn't change the probability of heads on the second. So P(two heads) = P(first head) × P(second head) = ½ × ½ = ¼.

Consider this Nigerian scenario: the probability of selecting a red ball from a bag is ⅓ and a blue ball is ⅙. The probability of selecting either red or blue is ⅓ + ⅙ = ½ (addition rule).

A probability distribution shows all possible outcomes of a random event and how likely each outcome is to happen. Think of it like listing every possible result when you roll a die, along with the chance of getting each number.

Imagine a Lagos lottery where 1000 tickets are sold and 50 winners are drawn. A probability distribution would tell you: the chance of winning one prize is 50/1000, winning nothing is 950/1000, and so on. This helps predict what typically happens over many draws.

There are two main types: discrete distributions (like coin tosses) where outcomes are distinct, and continuous distributions (like heights of students) where outcomes form a range. Most WAEC questions focus on discrete distributions because they're easier to calculate.

Understanding distributions helps you solve real problems about games, business decisions, and chance events you encounter daily.

A scalar quantity is something that has only size or magnitude, like when you say the temperature in Lagos is 28°C or a book costs ₦2,500. Magnitude alone tells the complete story. A vector quantity, however, needs both magnitude and direction to be fully described. For example, if someone asks where Abuja is from your location, saying "200 kilometres" isn't enough—you must add direction like "200 kilometres northeast" for complete information.

Think of it this way: distance is scalar (just the amount travelled), but displacement is vector (amount and direction from starting point). Wind speed of 15 km/h is scalar, but wind blowing at 15 km/h from the north is vector. Speed is scalar; velocity is vector.

A vector is a quantity that has both magnitude (size) and direction, unlike scalars which only have magnitude. Think of it like giving someone directions to Lekki from Lagos Island – you must specify how far (magnitude) and which way (direction).

Vectors can be represented in three main ways. First, as arrows on a diagram where length shows magnitude and the arrowhead shows direction. Second, using column notation like (3, 4) or in three dimensions (2, 5, 1). Third, using the form xi + yj + zk where i, j, and k are unit vectors along the x, y, and z axes respectively.

Consider a trader in Balogun Market pushing a cart northeast with force 50N. The force is a vector because it has both the 50N strength and the northeast direction. You can write this as components: how much force acts east and how much acts north.