Math  /  Data & Statistics

QuestionCorrect
Carter has 213 songs on a playlist. He's categorized them in the following manner: 11 gospel, 28 pop, 36 rock, 19 classical, 23 country, 43 folk, and 53 jazz. If Carter begins listening to his playlist on shuffle, what is the probability that the first song played is a pop song? Express your answer as a fraction in lowest terms or a decimal rounded to the nearest millionth.

Studdy Solution

STEP 1

What is this asking? Out of all of Carter's songs, what are the chances the first random song is pop? Watch out! Don't forget to simplify your fraction at the end!

STEP 2

1. Total Songs
2. Pop Songs
3. Probability

STEP 3

Let's **add** all the different types of songs together to find the **total** number of songs Carter has.
This is important because we need to know the **whole** to find the probability of a **part**!

STEP 4

11+28+36+19+23+43+53=21311 + 28 + 36 + 19 + 23 + 43 + 53 = 213 So, Carter has a **whopping** 213213 songs!

STEP 5

We're told directly that Carter has 2828 pop songs.
That's **fantastic**!
This is the number of "successful" outcomes we're looking for.

STEP 6

Probability is all about finding the chance of something happening.
It's a fraction where the **top** is the number of ways we can get what we want (**pop songs**) and the **bottom** is the total number of possibilities (**all songs**).

STEP 7

We know Carter has 2828 pop songs, and a total of 213213 songs.
So, the probability of hearing a pop song first is: 28213\frac{28}{213}

STEP 8

Now, let's see if we can simplify this fraction.
Are there any common factors between 2828 and 213213?
Well, 2828 is 474 \cdot 7, and 213213 is 3713 \cdot 71.
Since there are no common factors other than 11, the fraction 28213\frac{28}{213} is already in its **simplest form**! **Awesome**!

STEP 9

If we want to express this probability as a decimal rounded to the nearest millionth, we can **divide** 2828 by 213213: 28÷2130.13145539928 \div 213 \approx 0.131455399 Rounded to the nearest millionth, this is 0.1314550.131455.

STEP 10

The probability that the first song played is a pop song is 28213\frac{28}{213}, or approximately 0.1314550.131455.

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