Question

A shopper buys 36 items at random in a supermarket where, because of the sales tax imposed, the final digit (the number of pence) in the price is uniformly and randomly distributed from 0 to \(9 .\) Instead of adding up the bill exactly she rounds each item to the nearest 10 pence, rounding up or down with equal probability if the price ends in a ' 5 '. Should she suspect a mistake if the cashier asks her for 23 pence more than she estimated?

Short answer

The expected error is close, and 23 pence is within likely variance. No mistake is suspected.

Step-by-step solution

Understanding the Problem

The problem involves rounding and random distribution. Each item price's final digit is uniformly distributed between 0 and 9 due to sales tax.

Expected Rounding Error

Calculate the expected rounding error when rounding each item to the nearest 10 pence.

Error Distribution Calculation

Since the price ends between 0 and 9, the average rounding error can be calculated for each digit.

Compute the Expected Total Error

Multiply the average error per item by the number of items to get the expected total rounding error.

Compare Expected and Actual Errors

Compare the expected total rounding error to the actual error of 23 pence.

Conclusion

Determine if the discrepancy between estimated and actual price is significant enough to suspect a mistake.

Key concepts

uniform distribution

In probability and statistics, uniform distribution is a type of distribution where all outcomes are equally likely. Imagine spinning a wheel with numbers 0 to 9; every number has an equal chance of being picked, 1 in 10, or 10%. In our exercise, the final digit of each item's price in the supermarket is uniformly distributed between 0 and 9 thanks to the sales tax. So, if you're looking at the last digit of any item price, you can expect it to be any number from 0 to 9 with equal probability. This even spread is what we call a uniform distribution. When rounding prices to the nearest 10 pence, this uniform distribution is essential for calculating the rounding errors accurately because each digit influences the error in a balanced way.

expected value

The expected value in a probability context is like a weighted average of all possible outcomes. It gives us an idea about the 'center' of a distribution. In our exercise, we're interested in the expected rounding error every time the shopper rounds a price digit to the nearest 10 pence. Let's break it down:

• Digits 0-4 round down to the nearest 10 pence (0), resulting in errors of 0, -1, -2, -3, and -4 pence respectively.
• Digits 6-9 round up to the nearest 10 pence (10), resulting in errors of +4, +3, +2, and +1 pence respectively.
• If the digit is 5, it could round either up or down with equal probabilities, leading to either +5 or -5 pence, averaging to 0 pence.

To find the expected rounding error for a single item, we calculate the average of these errors weighted by their probability (which, due to uniform distribution, is the same for each digit). This helps in understanding how much error, on average, we expect per item, which is crucial before scaling up to multiple items.

random rounding errors

Random rounding errors happen due to the act of rounding numbers to a certain precision. These small deviations can add up to a noticeable difference when you deal with many items, like our shopper's cart of 36 items. When prices have random final digits, and you round these to the nearest 10 pence, every rounding decision introduces a small error. Since these errors can be positive or negative, you might think they would cancel each other out. However, due to the structure of our specific problem, you're likely to have a more consistent pattern. The uniform distribution helps predict the kind of errors we will encounter, but remember: randomness implies unpredictability in smaller samples. This is why we do an expected value calculation—you get an average idea, but not a precise prediction.

price estimation

When we estimate prices, especially using approximations like rounding, our goal is to be close enough to the actual total. For our shopper rounding the prices of 36 items, the small rounding errors on each item might seem insignificant individually, but they accumulate. Her estimate is based on these approximations.

To determine if the cashier's demand for 23 pence more than her estimate is reasonable, she needs to understand the expected total rounding error. By multiplying the expected error per item by the number of items, she gets an expected total error she can compare with the cashier's figure. If the discrepancy of 23 pence is significantly larger than what the expected error calculates, she might suspect a mistake. This method balances practical estimation with statistical reliability, using expected values and uniform distribution principles to ground her estimate in solid math.