Question

1. carlas music playlist has 80 rock songs, 40 jazz songs, 25 country songs, 30 hip - hop songs, and 45 classical music songs. which statements are true? i. if carla plays 25 songs at random, she should expect about 5 country music songs. ii. if carla plays songs at random, she should expect less than half of them to be jazz or classical music songs. iii. if carla plays 100 songs at random, she should expect about 13 hip - hop songs. clear all

Explanation:

Step1: Calculate total number of songs

The total number of songs in the playlist is \(80 + 40+25 + 30+45=220\) songs.

Step2: Calculate expected - value for statement I

The probability of a country - music song is \(P(\text{country})=\frac{25}{220}\). If she plays \(25\) songs, the expected number of country - music songs is \(E_1 = 25\times\frac{25}{220}=\frac{625}{220}\approx2.84\lt5\). So statement I is true.

Step3: Calculate expected - value for statement II

The probability of a jazz song is \(P(\text{jazz})=\frac{40}{220}\). If she plays \(45\) songs, the expected number of jazz songs is \(E_2 = 45\times\frac{40}{220}=\frac{1800}{220}\approx8.18\gt\frac{45}{2} = 22.5\) is false.

Step4: Calculate expected - value for statement III

The probability of a hip - hop song is \(P(\text{hip - hop})=\frac{30}{220}\). If she plays \(100\) songs, the expected number of hip - hop songs is \(E_3=100\times\frac{30}{220}=\frac{3000}{220}\approx13.64\approx13\). So statement III is true.

Answer:

I and III only