Lesson 80: Calculating Frequency Distributions

New Concept

A frequency distribution shows the number of observations falling into several ranges of data values.

What’s next

Next, you’ll use tables and graphs to build your own frequency distributions and calculate the probability of different outcomes.

Property

A compound event is an event consisting of two or more simple events.

Explanation

Think of this as a combo deal! A simple event is like getting just fries. A compound event is getting a burger AND fries. It’s multiple simple things happening together, like flipping a coin while also rolling a die. We then look at all the awesome combinations that can happen from that one experiment.

Examples

• Flipping a coin and rolling a die can result in a combined outcome like $(\text{Heads}, 5)$.
• Spinning a spinner with {Red, Blue} sections and flipping a coin gives outcomes like $(\text{Red, Tails})$.
• Drawing two cards from a standard deck can result in a compound outcome like getting a $(\text{King of Hearts}, \text{Ace of Spades})$.

What happens when we combine two simple events? Let's map out all the possibilities for a compound event, our second key idea.

Example Problem
A student spins a spinner and flips a fair coin. Find the theoretical probability of each outcome.
The spinner has five equal sections: Red, Blue, Yellow, Red, Green.

Step-by-Step

1. We need to create a table to show all possible outcomes. The column headings will be the colors from the spinner. Since 'Red' appears twice, we'll have two 'Red' columns. The row headings will be 'Heads' and 'Tails' for the coin flip.

Property

A frequency distribution shows the number of observations falling into several ranges of data values. Tables, graphs, tree diagrams, and lists are used to show frequency distributions.

Explanation

Ever wonder which number is the luckiest when you roll dice? A frequency distribution is the ultimate scoreboard for your data! It neatly organizes results to show how often each outcome pops up. We use cool tools like bar graphs or simple tables to quickly see which results are the rock stars and which are just rare cameos.

Examples

• For the word "BOOKKEEPER", a frequency table shows the count for each letter: {B:1, O:2, K:2, E:3, P:1, R:1}.
• If you roll two dice 40 times, a frequency table might show the sum of '7' occurred 8 times, while the sum of '2' occurred only once.
• A survey of 50 students' favorite pets could be shown in a bar graph: {Dogs: 25, Cats: 15, Fish: 7, Other: 3}.

Let's see how a player's performance can be broken down using a frequency distribution, which is our first key idea.

Example Problem
A basketball player's shots are recorded. Graph the frequency distribution and find the experimental probability of each outcome.

Property

Probability is the ratio of desired results to all possible results:

Explanation

Probability is just a cool mathematical way of asking, "What are the chances?" To figure it out, you count the number of ways you can win and divide it by the total number of things that could possibly happen. This simple fraction tells you exactly how likely you are to get the result you want, from totally impossible to absolutely certain!

Examples

• The probability of rolling a 5 on a standard six-sided die is $P(5) = \frac{1}{6}$, since there's one '5' and six total sides.
• In a bag with 4 red and 6 blue marbles, the probability of drawing a red marble is $P(\text{red}) = \frac{4}{10} = \frac{2}{5}$.
• For the word "MISSISSIPPI", the probability of drawing an 'S' is $P(S) = \frac{4}{11}$ because there are 4 'S' letters out of 11 total.