## The iPhone Epidemic

Apple released its new line in the iPhone family back in mid-September with the iPhone 5. Many critics and blog reviews (like http://www.techradar.com/reviews/phones/mobile-phones/iphone-5-1096004/review/1#articleContent) had many good and bad things to say about Apple’s newest product. Apple has definitely made an improvement from the previously released 4S with a lighter and dimensionally larger phone as well as a more aesthetic device with a smooth two-toned backing along with a slightly larger screen. The new iPhone also allows for 4G coverage, increasing its overall speed from the 4S. However there are some downs to this new product. Firstly it is expensive (~$200) and on top of that it needs all new adapters that go for $30 each, therefore it is not very compatible with past iPhone users. Also its design isn’t ideal or very durable. The metal edging around its circumference flakes easily just from pocket wear and the headphone jack is in a less convenient location. With all that being said the iPhone 5 is an upgrade from its predecessor however as far as compared to all other smartphones it could have been better. Others screens are larger, have longer lasting batteries, a more advanced operating system, and are less expensive.

Even with its less than jaw dropping innovations the iPhone 5 and Apple are still one of the most sold phones in the world. Looking at Information-Based Imitation and by using Conditional Probability one can examine this market trend.

Phones seem to be very seasonal. I can remember when just having a cell phone was cool, no matter what it was. This then evolved into the craze for flip-phones, blackberries, and most recently iPhones. All these phone fads are examples of Information-Based Imitation. When one needs a new phone and are deciding on what phone to buy they take into account two things: what do I, personally, know about this product and what is everyone else buying. They have definitely heard good and bad things about the product, either from word of mouth or through some sort of review. This is all private information, information that they have amassed one the product that could aid in their decision making process. Despite what one individual knows privately they almost always take into consideration what other people are doing. This is an example of one seeing the decisions that other people have made without knowing any of the information that they may have known when they made a decision. Obtaining private knowledge and seeing others’ actions without seeing the information they based those action on are the basic ingredients to form an Information-Based Imitation.

A mathematical expression for Info-Based Imitation can be seen by using Conditional Probability, essentially the mathematical thought process on all individuals that are looking to purchase a new phone. As an example I with calculate the probability that A, you should get an iPhone, given B, that you need a new phone ( Pr[A/B] ). Using Conditional Probability one can write out the following formula using Bayes’ Rule:

Using the date from http://www.tech.sc/npd-announces-that-android-surpassed-ios-in-contradiction-with-the-carriers-reports/ the probability that one buys an iPhone is approximately .30. To get the probability one of needing a new phone we can evaluate the following expression:

Pr[B] = Pr[A] * Pr[B/A] + Pr[notA] * Pr[B/notA]

We know that Pr[A] is .30 therefore the Pr[notA] must be .70. Now assuming of probability that you need a new phone given that you bought an iPhone ( Pr[B/A] ) to be .05 and the probability that you need a new phone given that you bought any other phone ( Pr[B/notA] ) to be .03 we discover that the probability of one needing a new phone is .036. Now plugging back into the formula for Bayes’ Rule we can find that the probability that someone buys an iPhone given that they need a new phone is about 42%. From this analysis we can see why many people find themselves buying the iPhone and why a large majority of people own the iPhone.

-Hemi7k77