Practise defining binomial random variables and justifying the model through a fixed trial count, binary outcomes, independence, and constant success probability.
Ms. Fey is a manager at a restaurant. To improve the dining experience for her customers, she uses a digital music service to create a playlist of songs that will be played in the restaurant.
The playlist contains 1,000 songs and consists of four different types of music in the following quantities: 200 country songs, 400 pop songs, 100 rock songs, and 300 jazz songs. The digital music service will select songs at random from the playlist to be played in the restaurant. Any song can be replayed at any time.
In every one-hour period, 20 songs will be played at random and any song can be replayed at any time. Ms. Fey is interested in how many rock songs will be played in a typical one-hour period.
Define the random variable of interest to Ms. Fey, and state how the random variable is distributed.
Model Solution
Let X be the number of rock songs among the 20 songs played in one hour. Since songs are selected independently with replacement and p=100/1000=0.10, X has a binomial distribution with n=20 and p=0.10.
Scoring components
1. Defines the random variable as the number of rock songs in one hour.
2. Identifies a binomial distribution.
3. States n=20 and p=0.10.