To find the probability that the first $8$ songs in played in Kenneth's playlist are country songs, we need to find divide the number of successes and divide it by the number of possible outcomes. Let's find the number of possible outcomes first!

Number of Possible Outcomes

Since the order of the songs does not matter, we will find the combinations of

n

objects given

r

at a time.

C (n, r) = \frac{n !}{( n - r ) ! r !}

The total number of possible outcomes are the combinations of

18

songs taken

8

at a time. Let's substitute these values into the formula and simplify!

C (n, r) = \frac{n !}{( n - r ) ! r !}

SubstituteII

$n = 18$ , $r = 8$

C (18, 8) = \frac{1 8 !}{( 1 8 - 8 ) ! 8 !}

SubTerm

Subtract term

C (18, 8) = \frac{1 8 !}{1 0 ! 8 !}

Write as a product

C (18, 8) = \frac{1 0 ! \cdot 1 1 \cdot 1 2 \cdot 1 3 \cdot 1 4 \cdot 1 5 \cdot 1 6 \cdot 1 7 \cdot 1 8}{1 0 ! 8 !}

CancelCommonFac

Cancel out common factors

C (18, 8) = \frac{1 0 ! \cdot 1 1 \cdot 1 2 \cdot 1 3 \cdot 1 4 \cdot 1 5 \cdot 1 6 \cdot 1 7 \cdot 1 8}{1 0 ! 8 !}

SimpQuot

Simplify quotient

C (18, 8) = \frac{1 1 \cdot 1 2 \cdot 1 3 \cdot 1 4 \cdot 1 5 \cdot 1 6 \cdot 1 7 \cdot 1 8}{8 !}

UseCalc

Use a calculator

C (18, 8) = 43758

There are

43758

posible outcomes.

Number of Successes

We know that there are $8$ country songs. Since the order of the songs does not matter, there is just a single combination that has all country songs. There is $1$ success.

Finding the Probability

Finally, we can find the probability by dividing the number of successes by the number of possible outcomes. Let's do it!

P = \frac{Number of successes}{Number of possible outcomes} ⇓ P = \frac{1}{4 3 7 5 8}

Therefore, the probability that the first

8

songs are country songs is

\frac{1}{4 3 7 5 8} .

{{ 'ml-heading-exercise' | message }} {{ exercise.name }} {{ 'ml-label-page' | message }} {{ exercise.page }}

{{ 'ml-solutions-missing-heading' | message }}