Probability

Ever wondered if it'll rain tomorrow or if you'll win a game? That's probability! It's a math way to guess how likely things are to happen.

An outcome is one possible result of a random experiment.

All possible outcomes are the complete list of everything that could happen in a random experiment.

What are all the possible outcomes when you flip a coin?

What are all the possible outcomes when you roll a six-sided die?

An event is a set of outcomes from all possible outcomes.

In the experiment of rolling a die, find the outcomes that correspond to rolling a number greater than 4.

We often use words to talk about probability. If something will never happen, it's impossible. If it will definitely happen, it's certain. In between, we say things like 'likely,' '50-50 chance,' or 'unlikely.' We can line them up from least to most likely.

• Impossible: It can’t happen.
  Example: Riding a dinosaur.
• Less likely: It probably won’t happen.
  Example: Rolling a die and getting a 3.
• Even chance: It has the same chance to happen or not.
  Example: Tossing a coin and getting head.
• Most likely: It will probably happen.
  Example: Smiling at school today.
• Certain: It will happen.
  Example: The sun will rise tomorrow.

When you flip a coin, there are two possible outcomes: heads or tails. The chance of getting heads is the same as getting tails—it’s 1 out of 2! In math, we write:
This means heads will happen about half the time!

The probability of an event, written \(P(\text{event})\), is a number that tells us how likely the event is to happen. It's always between 0 (impossible) and 1 (certain).

Probabilities can be expressed in multiple forms, such as fractions, decimals, or percentages, depending on the context or preference. Each form provides the same information but in a different representation, making it easier to understand or compare likelihoods.

The probability of an event "even chance" can be represented as:

• Fraction: \( \frac{1}{2} \)
• Decimal: To convert the fraction to a decimal, divide the numerator by the denominator: \( 1 \div 2 = 0.5 \).
• Percentage: To convert the decimal to a percentage, multiply by 100: \( 0.5 \times 100 = 50\pourcent \).

In some experiments, every outcome is equally likely to occur, such as when flipping a fair coin or olling a fair die. These are referred to as equally likely outcomes.

When all outcomes are equally likely, the probability of an event is given by the formula:$$P(\text{event}) = \frac{\text{number of outcomes in the event}}{\text{total number of possible outcomes}}$$

Calculate the probability of rolling an even number with a fair six-sided die.

Isaac wishes to determine how a cone lands when tossed—base down or point down? The possible outcomes are as follows:

• Base down:
• Point down:

Due to the cone’s shape potentially favoring one outcome, Isaac cannot predict the probabilities. He conducts 50 trials and records the results:

• Base down: 30 times.
• Point down: 20 times.

The chance of landing base down is approximately 30 times over 50 times. So, he estimates:

• \( P(\text{"base down"}) = \frac{30}{50} = 0.6 \) (60\(\pourcent\)).
• \( P(\text{"point down"}) = \frac{20}{50} = 0.4 \) (40\(\pourcent\)).

The more trials he conducts, the closer his estimates approach the true probabilities.

The probability of an event, \( P(\text{event}) \), can be estimated using the formula:$$P(\text{event}) \approx \frac{\text{number of times the event occurs}}{\text{number of trials}}$$Here, "trials" refer to the number of times the experiment is repeated.