Problem 55 - 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. On a game show, you are given five digits to arrange in the proper order to form the price of a car. If you are correct, you win the car. What is the probability of winning, given the following conditions? (a) You guess the position of each digit. (b) You know the first digit and guess the positions of the other digits.

Step-by-Step Solution

Answer

The probability of guessing the car's price correctly when all positions are guessed is \(\frac{1}{120}\). When the first digit's position is known, the probability of guessing correctly increases to \(\frac{1}{24}.\)

Step 1: Determine the total number of possibilities

There are 5 digits and each digit can be placed in any of the 5 positions. Using the principle of counting, we know that there are \(5!\) (factorial of 5) ways of arranging these 5 digits. This gives us a total of \(5*4*3*2*1 = 120\) possible arrangements.

Step 2: Calculate the probability of guessing correctly (Scenario A)

In this scenario, all positions of the digits are guessed. Since there's only 1 correct arrangement of the digits, the probability of guessing it correctly is then \(\frac{1}{total \ number \ of \ possibilities}\) which is \(\frac{1}{120}.\)

Step 3: Calculate the probability of guessing correctly (Scenario B)

In this scenario, the first digit's position is known. Now, instead of 5 positions to guess, there are only 4 left for the remaining digits. The total possible arrangements now are \(4!\) which is \(4*3*2*1 = 24\). Since there's only one correct arrangement, the probability of guessing correctly in this case is \(\frac{1}{24}\).

Key Concepts

Permutation

When you have several items and need to arrange them, you're dealing with permutations. Permutations refer to the different ways of arranging a set number of items. For example, arranging the digits to form the price of a car involves permutations. In our game show question, if you permute 5 unique digits, each arrangement is a permutation. What's fascinating is how each change in arrangement forms a new sequence or order. When all items are unique, the calculation of permutations is straightforward—use a factorial. This allows us to count all possible orders without repetition. Permutations are crucial in problems where the order matters, just like the order of guessing digits correctly in our problem. For instance, with our 5 digits, every unique order changes the number formed, hence a new permutation.

Factorial

A factorial, often denoted by an exclamation mark, is a mathematical operation.It represents the product of an integer and all the integers below it. For instance, the factorial of 5, written as \(5!\), is calculated by multiplying 5 by 4, by 3, down to 1:\[5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\].The concept of factorial is vital for counting permutations and combinations.In our exercise, calculating \(5!\)gave us the total number of all possible arrangements of 5 digits.Factorial grows rapidly; even a small increase in the integer results in significantly larger numbers.This property makes it perfect for calculating large sample spaces as seen in counting problems.Whenever you're arranging 'n' different items, \(n!\)is your go-to formula.

Sample Space

In probability, a sample space is the set of all possible outcomes.For the car pricing problem, each unique arrangement of the digits forms part of the sample space.When you hear 'sample space,' think about all the different results that could happen.In scenario A, with no digits fixed, the sample space was 120 arrangements, produced by \(5!\).Each possibility represents a different way to arrange the car price digits.For scenario B, knowing the first digit reduces our sample space dramatically.Now we explore only 24 possibilities, as given by \(4!\).Knowing some information ahead of time helps limit the sample space, which can make it easier to find successful outcomes.Understanding sample space is crucial as it forms the basis of calculating probabilities for different outcomes in many scenarios.