# Number of Glasses in the Tray: Solving the Probability Puzzle

Probability is a fascinating field of mathematics that deals with the likelihood of occurrence of a particular event. It has a wide range of applications, from predicting weather patterns to making strategic decisions in business. One such intriguing probability puzzle is the “Number of Glasses in the Tray” problem. This problem involves a tray with a certain number of bottles and glasses, and the probability of randomly picking a bottle from the tray. The puzzle is to find out the number of glasses in the tray, given certain conditions. Let’s delve into this problem and understand how to solve it.

## Understanding the Problem

The problem statement is as follows: “There are x bottles and y glasses in a tray and the probability of randomly picking a bottle is 2/5. 4 bottles are added to the tray and the probability of picking a bottle becomes 4/7. What is the number of glasses in the tray?”

## Breaking Down the Problem

Before we start solving the problem, let’s break it down into smaller parts. The first part of the problem tells us that the probability of picking a bottle from the tray is 2/5. This means that the ratio of the number of bottles to the total number of items in the tray (bottles + glasses) is 2:5. The second part of the problem tells us that when 4 more bottles are added to the tray, the probability of picking a bottle becomes 4/7. This means that the ratio of the number of bottles (x + 4) to the total number of items in the tray (bottles + glasses + 4) is 4:7.

## Solving the Problem

From the first part of the problem, we can write the equation x/(x+y) = 2/5. Solving this equation for y, we get y = 1.5x. From the second part of the problem, we can write the equation (x+4)/(x+y+4) = 4/7. Substituting y = 1.5x into this equation, we get (x+4)/(2.5x+4) = 4/7. Solving this equation for x, we get x = 8. Substituting x = 8 into the equation y = 1.5x, we get y = 12. Therefore, the number of glasses in the tray is 12.

## Conclusion

The “Number of Glasses in the Tray” problem is a classic example of a probability puzzle that can be solved using basic algebra. The key to solving such problems is to break down the problem into smaller parts and translate the words into mathematical equations. Once the equations are set up, the problem becomes a simple matter of solving the equations to find the solution.