Problem 60 - Precalculus with Limits

Question

In Exercises 53 - 60, the sample spaces are large and you should use the counting principles discussed in Section 9.6. TM personal identification number(PIN) codes typically consist of four-digit sequences of numbers. Find the probability that if you forget your PIN,you can guess the correct sequence (a) at random and (b) if you recall the first two digits.

Step-by-Step Solution

Answer

The probability of guessing a four-digit PIN correctly at random is \( \frac{1}{10000} \), and the probability of guessing correctly when remembering the first two digits is \( \frac{1}{100} \).

Step 1: Calculating the Total Number of Outcomes

The first step is to calculate the total number of possible outcomes. Since each digit of the PIN can be from 0 to 9, there are 10 possibilities for each digit. Therefore, the total number of possible four-digit codes is \(10^4\) or 10000.

Step 2: Calculating the Probability of Guessing Correctly at Random

The probability of guessing correctly at random is the number of successful outcomes divided by the total number of outcomes. In this case, there is only one successful outcome (guessing the right PIN), and 10000 total possible outcomes. Therefore, the probability \( P(a) \) is \( \frac{1}{10000} \).

Step 3: Calculating the Probability of Guessing Correctly when Remembering the First Two Digits

If the first two digits of the PIN are remembered, there are only two digits that need to be guessed. Each digit still has 10 possible values, so the total number of possible PINs is now \(10^2\) or 100. The probability \( P(b) \) of guessing correctly is again one successful outcome divided by the total number of outcomes, so \( P(b) = \frac{1}{100} \).

Key Concepts

Probability

Probability is a mathematical concept that measures the likelihood of a certain event happening. It's expressed as a number between 0 (impossible event) and 1 (certain event). When you calculate probability, you compare the number of favorable outcomes to the total number of possible outcomes. For example, if there is only one correct PIN out of 10,000 possible combinations, the probability of guessing it correctly by chance is quite low.

• If an event is certain, its probability is 1.
• If an event is impossible, its probability is 0.
• An event with equal chances of happening and not happening has a probability of 0.5.

Remember that probability can also be expressed as a percentage. In the case of the ATM PIN, \(P(a) = \frac{1}{10000} = 0.0001\), which is 0.01%.

Sample Space

A sample space is the set of all possible outcomes of a particular experiment or activity. For instance, when considering an ATM PIN that is made up of four digits—from 0 to 9—each digit can independently take on any of these 10 values.Therefore, to figure out the total sample space for a four-digit PIN, you multiply the number of possible values for each digit. This means the entire sample space is \(10 \times 10 \times 10 \times 10 = 10000\). Keeping track of the sample space is vital, because it forms the basis on which we calculate probabilities.

Permutations

Permutations come into play when the order of objects is important. It's a concept in counting principles where you consider the possible arrangements of a set of items. For ATM PINs, however, each digit is independent, and we're not scrambling them into different orders, so permutations as such aren't directly applicable to our current problem. But if you had a PIN where the sequence mattered beyond simple digs, you would need to calculate permutations, which can be tricky for large sets.Permutations formula is given by \(nPr = \frac{n!}{(n-r)!}\), where \(n\) is total items to choose from and \(r\) is items to arrange.

Combinations

Combinations are about selecting items without concern for the order in which they are picked. This is different from permutations, where order does matter. For example, if you were combining different elements of a password but didn't care in what order they appeared, you would use combinations.In the context of guessing ATM PINs, if ever you wanted to allow order variations or repetitions, combinations would be relevant. Typically, combinations are computed using the formula:\(nCr = \frac{n!}{r!(n-r)!}\).Although our example involves guessing a four-digit code, combinations aren't necessary since every digit's order matters to match the specific PIN.