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Chapter 7: Problem 42

Assume that you already know the first digit of the combination for the lock described in Exercise 41 . Find the probability that a random guess of the remaining three digits will open the lock.

Short Answer

Expert verified
The probability of guessing the remaining three digits correctly in a random attempt is \(\frac{1}{1000}\) or 0.001.

Step by step solution

01

Determine the number of possible outcomes for each digit

Since we are dealing with a combination lock that uses digits 0-9 for each digit, there are 10 possible outcomes for each remaining digit (1st digit is already known).
02

Calculate the probability of guessing each digit correctly

Since there are 10 possible outcomes for each digit, the probability of guessing a single digit correctly is 1/10 or 0.1.
03

Use the rule of product for independent events to find the probability of guessing all digits correctly

To find the probability of guessing all three digits correctly, we will multiply the probabilities of guessing each individual digit. This will give us the probability of guessing all the digits correctly in a single attempt. Probability of guessing all 3 digits correctly = (Probability of guessing 2nd digit correctly) x (Probability of guessing 3rd digit correctly) x (Probability of guessing 4th digit correctly)
04

Calculate the probability

Using the probabilities found in Step 2, let's calculate the probability of guessing all three digits correctly in a single attempt: Probability of guessing all 3 digits correctly = (1/10) x (1/10) x (1/10) = 1/1000 The probability of guessing the remaining three digits correctly in a random attempt is 1/1000 or 0.001.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Combination lock
A combination lock is a locking mechanism that uses a sequence of numbers or symbols as the combination to open it. These locks are commonly used for securing lockers, safes, or bicycles. Typically, combination locks utilize a dial or a set of rotating discs with numbers 0 through 9.
  • In the case discussed here, the lock requires a sequence of four digits to open.
  • Each digit in the sequence can range from 0 to 9, giving 10 possibilities for each.
  • The exercise assumes we know the first digit, reducing uncertainty to three digits.
Understanding how many possible combinations there are for a lock is foundational before solving probability problems.
Independent events
Independent events are scenarios in which the outcome of one event does not affect the outcome of another.

This concept is crucial when dealing with probabilities in combination locks, as choosing one digit of the combination does not alter the options available for the subsequent digits.
  • For instance, if you already know the first digit, guessing the second digit correctly does not change the probability of guessing the third or fourth digit correctly.
  • Each digit selection is an independent event with no influence over any other digit.
Being aware of this concept helps clarify why we can simply multiply probabilities to determine the combined chance of multiple correct guesses.
Rule of product
The rule of product is also known as the multiplication principle. This rule states that if there are multiple stages or components to an event, and each stage is independent, the total number of outcomes can be found by multiplying the number of options at each stage.

In probability terms, this helps us calculate the probability of several independent events occurring together.
  • For example, knowing there are 10 possible choices for each of the three unknown digits of the combination lock, we multiply these together to find the total number of possible combinations, which is 10 x 10 x 10 = 1000.
  • The probability of any single trial succeeding, or picking the correct combination, is found by taking the reciprocal of the total possible combinations, resulting in a probability of 1/1000 for this lock problem.
This rule is fundamental in assessing probabilities when dealing with combination locks or similar problems.
Random guess
Making a random guess means selecting an option purely by chance, without any influence or strategy. In probability exercises, assuming a random guess often helps in understanding outcomes without biases or prior knowledge.

When applied to combination locks, a random guess implies trying to open the lock without information other than what's already given.
  • Since we know the first digit, a random guess involves correctly identifying the remaining three digits.
  • As determined, the chance for each digit to be correct is 1 out of 10.
  • Combining these guesses for three digits follows the calculated probability of 1/1000.
Emphasizing the random nature of guessing highlights the role of probability in predicting outcomes in unpredictable situations.

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Most popular questions from this chapter

If \(B \subseteq A\) and \(P(B) \neq 0\), why is \(P(A \mid B)=1 ?\)

Other dice are weighted so that each of \(2,3,4\), and 5 is half as likely to come up as 1 is, and 1 and 6 are equally likely.

According to a study conducted by the Harvard School of Public Health, a child seated in the front seat who was wearing a seatbelt was \(31 \%\) more likely to be killed in an accident if the car had an air bag that deployed than if it did not. \({ }^{50}\) Let the sample space \(S\) be the set of all accidents involving a child seated in the front seat wearing a seatbelt. Let \(K\) be the event that the child was killed and let \(D\) be the event that the airbag deployed. Fill in the missing terms and quantities: \(P(\longrightarrow \mid \longrightarrow)=\longrightarrow P(\longrightarrow \mid \longrightarrow)\). HINT [When we say "A is \(31 \%\) more likely than \(\mathrm{B}\) " we mean that the probability of \(\mathrm{A}\) is \(1.31\) times the probability of B.]

Auto Sales In April 2008, the probability that a randomly chosen new automobile was manufactured by Ford was \(.15\), while the probability that it was manufactured by Chrysler was .12. \({ }^{35}\) What is the probability that a randomly chosen new automobile was manufactured by neither company?

Employment You have worked for the Department of Administrative Affairs (DAA) for 27 years, and you still have little or no idea exactly what your job entails. To make your life a little more interesting, you have decided on the following course of action. Every Friday afternoon, you will use your desktop computer to generate a random digit from 0 to 9 (inclusive). If the digit is a zero, you will immediately quit your job, never to return. Otherwise, you will return to work the following Monday. a. Use the states (1) employed by the DAA and (2) not employed by the DAA to set up a transition probability matrix \(P\) with decimal entries, and calculate \(P^{2}\) and \(P^{3}\). b. What is the probability that you will still be employed by the DAA after each of the next three weeks? c. What are your long-term prospects for employment at the DAA? HIIIT [See Example 5.]
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