Problem 41 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The following table, compiled in 2004 , gives the percentage of music downloaded from the United States and other countries by U.S. users: $$ \begin{array}{lcccccccc} \hline \text { Country } & \text { U.S. } & \text { Germany } & \text { Canada } & \text { Italy } & \text { U.K. } & \text { France } & \text { Japan } & \text { Other } \\ \hline \text { Percent } & 45.1 & 16.5 & 6.9 & 6.1 & 4.2 & 3.8 & 2.5 & 14.9 \\\ \hline \end{array} $$ a. Verify that the table does give a probability distribution for the experiment. b. What is the probability that a user who downloads music, selected at random, obtained it from either the United States or Canada? c. What is the probability that a U.S. user who downloads music, selected at random, does not obtain it from Italy, the United Kingdom (U.K.), or France?

Step-by-Step Solution

Answer

a. Yes, the table represents a probability distribution as the sum of all probabilities equals 100%. b. The probability that a user who downloads music, selected at random, obtained it from either the United States or Canada is 52.0%. c. The probability that a U.S. user who downloads music, selected at random, does not obtain it from Italy, the United Kingdom, or France is 85.9%.

Step 1: Verify probability distribution

To ensure that the given table represents a probability distribution, we must check if the sum of all probabilities adds up to 1. We will add all the percentages and check if the sum equals 100%. If the sum equals 100%, then the given table does represent a probability distribution. $$ 45.1\% + 16.5\% + 6.9\% + 6.1\% + 4.2\% + 3.8\% + 2.5\% + 14.9\% = 100\% $$ Since the sum of probabilities equals 100%, the table represents a probability distribution.

Step 2: Probability from United States or Canada

To find the probability that a randomly selected user downloaded music from either the United States or Canada, we will add the probabilities of music downloaded from both countries. $$ P(\text{U.S. or Canada}) = P(\text{U.S.}) + P(\text{Canada}) = 45.1\% + 6.9\% = 52.0\% $$ The probability that a user who downloads music, selected at random, obtained it from either the United States or Canada is 52.0%.

Step 3: Probability not from Italy, U.K., or France

To find the probability that a randomly selected U.S. user does not download music from Italy, the United Kingdom, or France, we will first find the probability of downloading music from these three countries and then subtract it from 100%. $$ P(\text{Italy, U.K., or France}) = P(\text{Italy}) + P(\text{U.K}) + P(\text{France}) = 6.1\% + 4.2\% + 3.8\% = 14.1\% $$ Now, we calculate the probability that a user does not download music from these three countries. $$ P(\text{Not Italy, U.K., or France}) = 100\% - P(\text{Italy, U.K., or France}) = 100\% - 14.1\% = 85.9\% $$ The probability that a U.S. user who downloads music, selected at random, does not obtain it from Italy, the United Kingdom, or France is 85.9%.

Key Concepts

Random Experiment Probability

When we talk about random experiments in probability, we're referring to any process or action that produces an outcome that cannot be predicted with certainty. Examples include flipping a coin, rolling a die, or, in this case, a user from the U.S. downloading music from various countries.

To characterize such activities, we use a probability distribution, which is a mathematical concept that assigns a probability to each possible outcome. These probabilities must add up to 100%, indicating total certainty that one of the outcomes will occur. In the music download statistics example, the table listed percentages which, when added together, equaled exactly 100%. This confirmed that the table provided a valid probability distribution for the music downloading experiment.

Music Download Statistics

Statistics often involve collecting, analyzing, and interpreting numerical data. When dealing with music download statistics, we're looking at data that may tell us, for example, how many people downloaded music within a certain period, and from which countries the music was obtained.

In our exercise, the data presented in the table showed the percentages of U.S. users downloading music from different countries. Interpreting this data provides insights into user preferences and market trends. For instance, 45.1% of the music was downloaded from the U.S., indicating a high preference for local music or possibly greater availability of U.S. music to the users.

Country-Based Download Probability

Country-based download probability gives us the likelihood that a person in the U.S. will download music from a particular country. This is important for understanding market dynamics and global music trends. From our exercise, we deduced that a random person in the U.S. had a 45.1% chance of downloading American music, and a 6.9% chance of downloading Canadian music.

To find the combined probability of a user downloading music from either the U.S. or Canada, we simply added the two individual probabilities, which is a fundamental rule in probability for mutually exclusive events—events that cannot happen simultaneously. The resulting 52% then represents our combined country-based probability for these two countries.

Complementary Probability

Complementary probability refers to the likelihood of an event not happening. It's a key concept in probability theory because it allows us to easily calculate the chances of an event by subtracting the probability of the event from 1 (or 100% in percentage terms).

In our exercise, we utilized complementary probability to determine the chance that a U.S. user did not download music from Italy, the U.K., or France. After finding the sum of the probabilities for these three countries, we subtracted it from 100% to obtain the complementary probability. This simple yet powerful concept is fundamental in various probability applications, making complex problems much more manageable.