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. Like when the weather app says there's a 90\(\pourcent\) chance of rain, that's probability telling us it's very likely. What else do you think we could use probability for?

Have you ever flipped a coin and wondered if it’d land on heads or tails? Or rolled a die and guessed what number you’d get? These are random experiments—things we do where the result isn’t certain until it happens.

An outcome is one possible result of a random experiment.

The sample space is the list of all possible outcomes of a random experiment.

What’s the sample space when you flip a coin?

What’s the sample space when you roll a six-sided die?

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

In math, we use capital letters like \(E\) to name events. So, we might say \(E\) is the event “it’s sunny tomorrow.”

You roll a die. Let \(E\) be the event of rolling an even number. What’s \(E\)?

The complementary event of an event \(E\) is everything in the sample space that isn’t in \(E\). We call it \(E'\) (\(E\)-prime).

You roll a die, and \(E\) is rolling an even number. What’s \(E'\)?

A multi-step random experiment is one that involves a sequence of actions, where each action (or step) has its own set of possible outcomes. In our example of tossing two coins, the experiment is multi-step because it involves two separate coin tosses:

• The first coin toss (step 1) can result in Heads (H) or Tails (T).
• The second coin toss (step 2) can also result in Heads (H) or Tails (T).

The overall outcome of the experiment is given by combining the outcomes of each step. For instance, the outcome \(HT\) means that the first coin landed on Heads and the second coin landed on Tails.Using different representations—such as grids, tables, tree diagrams, or simply listing the outcomes—helps us organize and visualize all the possible combinations that result from the multiple steps.

When an experiment involves more than one step, we can represent the sample space (the set of all possible outcomes) in several ways:

• using a grid (to visually map combinations along two axes),
• using a table (to organize outcomes in rows and columns),
• using a tree diagram (to follow each sequential step), or
• by listing all possible outcomes.

For the random experiment of tossing two coins, display the sample space by:

1. using a grid
2. using a table
3. using a tree diagram
4. listing all possible outcomes

The probability of an event \(E\), written \(P(E)\), tells us how likely \(E\) is to happen. It’s always a number between 0 and 1:

• 0 means impossible (0\(\pourcent\) chance).
• 1 means certain (100\(\pourcent\) chance).

For example, the probability the sun rises tomorrow morning is 1!

We can show probability as a fraction, decimal, or percentage. Like a 50-50 chance is \(\frac{1}{2}\), 0.5, or 50\(\pourcent\). Picture it on a number line: 0 is impossible, 1 is certain, and 0.5 is right in the middle!

What’s the probability the sun rises tomorrow morning?

Sometimes, every outcome in an experiment has the same chance—like flipping a fair coin or rolling a fair die. We call these equally likely outcomes.

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

What’s the probability of rolling an even number with a fair six-sided die?

For any event E and its complementary event E', $$P(E) + P(E') = 1 \quad \text{or} \quad P(E') = 1 - P(E).$$

Farid has a 0.8 (80\(\pourcent\)) chance of finishing his homework on time tonight (event \(E\)). What’s the chance he doesn’t finish on time?

Independent events are like you choosing an ice cream flavor and your friend picking a movie—your choice doesn’t affect theirs, and theirs doesn’t affect yours! For example, flipping a coin and rolling a die at the same time: whether you get heads or tails has no effect on what number you roll. Let’s explore this idea together!

Two events, \(A\) and \(B\), are independent if the chance of both happening is just the product of their individual chances. Mathematically:$$P(A \text{ and } B) = P(A) \times P(B)$$

Imagine you do two totally separate actions:

1. Flipping a fair coin (heads or tails).
2. Rolling a fair six-sided die (1, 2, 3, 4, 5, or 6).

What’s the probability of getting tails and rolling a number greater than 4 (like a 5 or 6)?

Let’s use a probability tree for our coin flip and die roll:

1. Draw the Tree: Start with two branches for the coin: “Heads” and “Tails.” Then, from each, draw two more branches for the die: “Number > 4” (5 or 6) and “Number \(\leqslant\) 4” (1, 2, 3, or 4).
2. Add Probabilities: Label each branch. The coin gives \(\dfrac{1}{2}\) for “Tails” and \(\dfrac{1}{2}\) for “Heads.” For the die, “Number > 4” is \(\dfrac{1}{3}\) (2 out of 6), and “Number \(\leqslant\) 4” is \(\dfrac{2}{3}\) (4 out of 6).
3. Follow the Path: To find \(P(\text{tails and number} > 4)\), trace the “Tails” branch, then the “Number > 4” branch, and multiply: $$\dfrac{1}{2} \times \dfrac{1}{3} = \dfrac{1}{6}$$
4. Wrap It Up: The tree confirms our answer—\(\dfrac{1}{6}\)!

Isaac wants to know how a cone lands when he tosses it—base down or point down? Here’s what it can do:

• Base down:
• Point down:

Since the cone’s shape might make one more likely, Isaac can’t guess. So, he tosses it 50 times and counts:

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

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 he tosses, the closer he gets to the real chances!

The probability \( P(E) \) of an event \( E \) can be estimated using the formula:$$P(E) \approx \frac{\text{number of times the event occurs}}{\text{number of trials}}$$