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 union of two events \(E\) and \(F\), denoted as \(E \text{ or } F\) or \(E \cup F\), is the event that occurs if either event \(E\) happens, event \(F\) happens, or both events happen simultaneously.

Consider the roll of a standard six-sided die.
Let event \(E\) be the event of rolling an even number. So, \(E = \{2, 4, 6\}\).
Let event \(F\) be the event of rolling a number less than 4. So, \(F = \{1, 2, 3\}\).
Find the event \(E\) or \(F\).

$$$$The intersection of two events \(E\) and \(F\), denoted as \(E \text{ and } F\) or \(E \cap F\), is the set of all outcomes that are common to both \(E\) and \(F\). The event \(E\) and \(F\) occurs if and only if both \(E\) and \(F\) happen simultaneously.

Consider the roll of a standard six-sided die.
Let event \(E\) be the event of rolling an odd number. So, \(E = \{1, 3, 5\}\).
Let event \(F\) be the event of rolling a number less than 4. So, \(F = \{1, 2, 3\}\).
Find the event \(E\) and \(F\).

Two events \(E\) and \(F\) are said to be mutually exclusive if they cannot occur simultaneously. In other words, the occurrence of event \(E\) excludes the possibility of event \(F\) occurring, and vice versa. This is mathematically represented as:$$E \text{ and } F = E \cap F = \emptyset,$$where \(\emptyset\) denotes the empty set, indicating that there are no common outcomes between \(E\) and \(F\).

Consider the roll of a standard six-sided die.
Let event \(E\) be the event of rolling an odd number. So, \(E = \{1, 3, 5\}\).
Let event \(F\) be the event of rolling an even number. So, \(F = \{2, 4, 6\}\).
Show that \(E\) and \(F\) are mutually exclusive.

Notation	Set Vocabulary	Probabilistic Vocabulary	Venn Diagram
\(U\)	Universal set	Sample space
\(x\)	Element of \(U\)	Outcome
\(\emptyset\)	Empty set	Impossible event
\(E\)	Subset of \(U\)	Event
\(x \in E\)	\(x\) is an element of \(E\)	\(x\) is an outcome of \(E\)
\(E'\)	Complement of \(E\) in \(U\)	Complement of \(E\) in \(U\)
\(E \text{ or } F\)	Union of \(E\) and \(F\): \(E \cup F\)	\(E\) or \(F\)
\(E \text{ and } F\)	Intersection of \(E\) and \(F\): \(E \cap F\)	\(E\) and \(F\)
\(E \cap F = \emptyset\)	\(E\) and \(F\) are disjoint	\(E\) and \(F\) are mutually exclusive

The probability of an event \(E\), denoted by \(P(E)\), is a value between 0 and 1 that represents the chance or likelihood of that event occurring.

• The probability of an impossible event is \(0\) or \(0\pourcent\).
• The probability of a certain event is \(1\) or \(100\pourcent\).
• The probability of any event is between \(0\) and \(1\), inclusive.

We can express probabilities using fractions, decimals, or percentages. For example, the probability of an event with a 50-50 chance can be written as \(\frac{1}{2}\), \(0.5\), or \(50\pourcent\).
We can visualize probabilities on a number line:The main goal of probability theory is to develop tools and techniques to calculate the probabilities of different events. Probability theory is built upon a set of axioms that form its foundation. Let us state and explain these axioms.

\(P\) is a probability if:

• \(0 \leqslant P(E) \leqslant 1\), for any event \(E\),
• \(P(U) = 1\),
• If \(E\) and \(F\) are mutually exclusive events, then \(P(E \text{ or } F) = P(E) + P(F)\).

• The probability of an event \(E\), \(\textcolor{colordef}{P(E)}\), is represented by the shaded area of the event in a Venn diagram:
• The first axiom states that the probability of an event \(E\) is a value between 0 and 1, inclusive.
• The second axiom states that the probability of the sample space \(U\) is equal to \(1\), i.e., \(100\pourcent\). This is because the sample space \(U\) includes all possible outcomes of a random experiment, so the event \(U\) always occurs, and \(P(U) = 1\). In a Venn diagram, this is represented as the entire shaded area of the sample space:$$\textcolor{colordef}{P(U)} = 1$$
• The third axiom states that if two events are mutually exclusive (i.e., they cannot occur simultaneously), then the probability of their union is the sum of their individual probabilities. In a Venn diagram, for two mutually exclusive events (with no overlap), the total area of their union is the sum of the individual areas:=$$\textcolor{olive}{P(E \text{ or } F)} = \textcolor{colordef}{P(E)} + \textcolor{colorprop}{P(F)}$$

For any event \(E\) with complementary event \(E'\),$$\textcolor{colordef}{P(E)} + \textcolor{colorprop}{P(E')} = 1$$Also,$$\textcolor{colorprop}{P(E')} = 1 - \textcolor{colordef}{P(E)}$$

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 ?

For any events \(E\) and \(F\),$$P(E \text{ or } F) = P(E) + P(F) - P(E \text{ and } F)$$

A local high school is holding a talent show where students can participate in singing, dancing, or both. The probability that a randomly selected student participates in singing is 0.4, the probability that a student participates in dancing is 0.3, and the probability that a student participates in both singing and dancing is 0.1. Find the probability that a randomly selected student participates in either singing or dancing.

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?

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:$$P(E) \approx \frac{\text{number of times the event occurs}}{\text{number of trials}}$$

When an experiment is performed \(n\) times (called trials) and the probability of an event occurring in each trial is \(p\), the expected value (or expectation) of the number of occurrences of that event is:$$n \times p$$This represents the average number of times the event is expected to occur over \(n\) trials.

Jane enters a raffle where the chance of winning a prize with each ticket is \(\frac{1}{10}\). She buys 20 raffle tickets. Find the expected number of prizes Jane is likely to win.