# Intro to Probability

Even though probability and statistics can be a challenging topic for many students, this video should provide you with a helpful foundation to tackle probability with confidence.

To begin, let’s think of simple probability as a type of counting problem. For example, if we roll a regular six-sided die and want to find the likelihood of rolling a “2,” we can determine that the probability will be \(\frac{1}{6}\) because we count 1 side on the die which has a “2” and 6 sides in total. In other words, there are 6 possible outcomes when we roll the die, but because we are only interested in one particular outcome, we count 1 in 6 as the probability of rolling a “2.”

Similarly, the probability of flipping a coin and getting “heads” is one-half, because we count 1 outcome of interest, “heads,” and 2 possible outcomes in total, “heads” and “tails.”

What if we wanted to find the probability of drawing an ace out of a shuffled deck of cards? Well, a regular deck of cards has 52 cards in total, and there are 4 aces in each deck (one ace for each suit). Because of this, the probability of drawing an ace from a shuffled deck of cards is 4 in 52, which can be simplified to 1 in 13.

When it comes to statistics and probability lessons, you’ll hear a number of special terms used for probability. Let’s go over some of the more common ones and what they mean.

In statistics, an **event** is one or more specific outcomes for an experiment or activity, such as flipping a coin or rolling a die. For example, rolling a “5” on a die is considered an event because it is one of the possible outcomes for rolling the die. We could also consider multiple outcomes to be an event, though, such as rolling an odd number. Events containing more than one single outcome are called **compound events**.

If we were interested in achieving a particular event in an experiment, this outcome would be called the **desired outcome**. Another way to refer to a desired outcome is a “**success**.” For example, in a game where players win by rolling a “6,” the outcome “6” on the die would be considered the desired outcome or success.

In statistics, you’ll sometimes come across independent and dependent events. Let’s look at examples of what those each mean. Let’s say I have four cards: the king and queen of spades, and the king and queen of hearts. Imagine that the cards are placed face-down and shuffled, and then one of the cards is chosen at random. What is the probability that the card would be a king? The probability would be \(\frac{1}{2}\) because there are 2 kings and 4 cards in all.

Now, assuming a king *had* been chosen, let’s consider what would happen if a second card is then chosen at random. The probability of choosing the second king would be \(\frac{1}{3}\), because there is only 1 king left and three cards on the table. This second pick is said to be done “without replacement” because we didn’t put the first king back on the table. In this case, the second event in the game would be classified as a **dependent event** because the probability of the second draw *depended* on the card was removed in the first draw.

If, however, the second pick had been done after putting the first king back on the table, the probability of drawing a king a second time would again be \(\frac{1}{2}\). This time, the second pick is done “with replacement” because the first selected card was placed back on the table. In this case, the game would be classified as two **independent events**, because the probability of getting a king on the second draw did not change based on whether a king was drawn on the first pick.

To keep dependent and independent events straight, ask yourself “does the probability of either event *depend* on the outcome of the other?” If it does, then the events are dependent. If not, then they are independent.

Let’s say we are rolling a six-sided die again. If we were interested in rolling a number between 1 and 6, we would say that there is a 100% success rate of that happening, because 1 through 6 are the *only* possible outcomes. This kind of event is called a **certain outcome** because it is sure to happen. On the other hand, if we wanted to roll a number 7 or larger on the six-sided die, this would be an **impossible outcome** because the die has no numbers larger than 6.

Suppose that while experimenting with rolling a die, we wanted to observe the event that the number 1 or an even number is rolled. In this case, we would be looking for the numbers 1, 2, 4, or 6. Because 1 itself is not an even number, though, we would say that 1 and the even numbers are two **mutually exclusive outcomes**—they have no points or events in common.

The last term we will discuss for this video is **random variables**. In statistics, we usually use the capital letter \(X\) to denote a random variable, and it is simply a placeholder for “the thing we are observing.” For example, when rolling a six-sided die, we would say that the random variable \(X\) describes the outcome of rolling the die. We can list out all the possible values that \(X\) can take using set notation: \(X = {1, 2, 3, 4, 5, 6}\). You may see something like this in your textbook as well:

While this may look a little funny at first, it is simply an abbreviated, mathematical way of saying “the probability that we observe the specific outcome little-\(x\).” We could use this to write the probability of rolling a “5” on a die as \(P(X=5)\), the probability that our random variable \(X\) (representing the die roll) has an outcome of 5.

With these definitions and concepts, you’re ready to try some problems on your own!

Thanks for watching, and happy studying!

## Frequently Asked Questions

#### Q

### What is probability?

#### A

Probability is the likelihood of a certain outcome occurring for a given event.

#### Q

### How do you calculate probability?

#### A

Calculate probability by dividing the number of acceptable outcomes by the number of possible outcomes.

\(P(A)=\frac{\text{# of acceptable outcomes}}{\text{# of possible outcomes}}\)

Ex. What is the probability of drawing a 4 from a standard deck of cards?

There are 4 fours in a deck of 52 cards, so \(P(A)=\frac{4}{52}=\frac{1}{13}\).

#### Q

### What is theoretical probability?

#### A

Theoretical probability is the likelihood of an event occurring. The formula for theoretical probability is:

\(P(A)=\frac{\text{number of acceptable outcomes}}{\text{number of possible outcomes}}\)

Ex. The probability of rolling a 2 on a standard 6-sided die is 1/6. There is one acceptable outcome and a total of six possible outcomes.

#### Q

### What is experimental probability?

#### A

Experimental probability is an estimate of the likelihood of a certain outcome based on repeated experiments or collected data.

Theoretical probability is based on what should happen, while experimental probability is based on what actually happened.

## Practice Questions

**Question #1:**

What is the probability of drawing a 7 from a standard deck of cards?

**Answer:**

There are 4 of each number in a standard deck of cards, one for each of the four suits. A standard deck of cards has 52 cards. Probability compares your desired outcome to the total outcome. Since there are 4 ways to get our desired outcome, and 52 total outcomes, the probability is \(\frac{4}{52}\). This fraction can be simplified by dividing both the numerator and denominator by 4: \(\frac{4\div4}{52\div4}=\frac{1}{13}\). Therefore, the probability of drawing a 7 from a standard deck of cards is \(\frac{1}{13}\).

**Question #2:**

What is the probability of rolling a prime number on a standard die?

**Answer:**

There are 6 numbers on a standard die. 3 of those numbers are prime: 2, 3, and 5. Since probability compares the desired outcome to the total number of outcomes, the probability is \(\frac{3}{6}\). This fraction can be simplified by dividing both the numerator and denominator by 3: \(\frac{3\div3}{6\div3}=\frac{1}{2}\). Therefore, the probability of rolling a prime number on a standard die is \(\frac{1}{2}\).

**Question #3:**

What is the probability of drawing a heart from a standard deck of cards?

**Answer:**

There are 4 suits in a standard deck of cards, with an even number of cards in each suit. Since we are wanting one particular suit, we compare this to the total number of suits to get the probability. Therefore, the probability of drawing a heart from a standard deck of cards is \(\frac{1}{4}\).

**Question #4:**

A baby shower party game involves drawing pacifiers out of a bag. The bag contains 15 blue pacifiers and 20 pink pacifiers. If you draw one pacifier out of the bag, what is the probability that it is a blue pacifier?

**Answer:**

Probability involves comparing the desired number of outcomes to the total number of outcomes. First, let’s find the total number of outcomes by adding together the number of blue and pink pacifiers. \(15+20=35\), so there are 35 total pacifiers in the bag. Since we want to know the probability of drawing a blue pacifier, we compare 15 to the total of 35: \(\frac{15}{35}\). This fraction can be simplified by dividing both the numerator and denominator by 5: \(\frac{15\div5}{35\div5}=\frac{3}{7}\). Therefore, the probability of drawing a blue pacifier out of the bag is \(\frac{3}{7}\).

**Question #5:**

To attract more tenants, an apartment complex paints the doors four different colors: red, blue, green, and yellow. There are 23 red doors in the complex, 19 blue doors, 17 green doors, and 21 yellow doors. If you are assigned an apartment at random, what is the probability that your door will be green?

**Answer:**

Probability involves comparing the desired number of outcomes to the total number of outcomes. First, let’s find the total number of outcomes by adding together the total number of doors: \(23+19+17+21=80\). Therefore, there are 80 possible outcomes. We are interested in the green doors, so there are 17 desired outcomes. We compare this to the total number of outcomes to get the probability. Therefore, the probability of being assigned a green door in the apartment complex is \(\frac{17}{80}\).