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Ch. 3 - Probability
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
All textbooksLarson 8th EditionCh. 3 - ProbabilityProblem 3.R.50
Chapter 3, Problem 3.R.50

In Exercises 49-53, use counting principles to find the probability.
50. A security code consists of three letters and one digit. The first letter cannot be A, B, or C. What is the probability of guessing the security code on the first try?

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Step 1: Determine the total number of possible outcomes for the security code. The code consists of three letters and one digit. For the first letter, there are 23 choices (since the first letter cannot be A, B, or C, leaving 26 - 3 = 23 letters). For the second and third letters, there are 26 choices each (no restrictions). For the digit, there are 10 choices (0 through 9). Multiply these possibilities together to find the total number of possible codes: 23 × 26 × 26 × 10.
Step 2: Recognize that there is only one correct security code. This means there is only one favorable outcome.
Step 3: Use the formula for probability: \( P = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}} \). Substitute the values from Step 1 and Step 2 into this formula.
Step 4: Simplify the fraction obtained in Step 3 to express the probability in its simplest form.
Step 5: Interpret the result as the probability of guessing the security code correctly on the first try.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Counting Principles

Counting principles, such as the multiplication and addition rules, are fundamental in determining the total number of possible outcomes in a scenario. In this case, we need to calculate the total combinations of letters and digits to find the probability of guessing the security code correctly.
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Fundamental Counting Principle

Probability

Probability is a measure of the likelihood of an event occurring, expressed as a ratio of favorable outcomes to the total number of possible outcomes. To find the probability of guessing the security code, we will divide the number of successful outcomes (correct code) by the total number of possible codes.
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Introduction to Probability

Permutations and Combinations

Permutations and combinations are methods used to count arrangements and selections of items. In this problem, we need to consider the restrictions on the first letter and how many choices we have for each position in the security code, which affects the total number of combinations.
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Permutations vs. Combinations
Related Practice
Textbook Question

In Exercises 17 and 18, use the table, which shows the numbers of first-time and repeat U.S. nursing students taking the National Council Licensure Examination (NCLEX-RN® exam) to pass or fail in a recent year. (Adapted from National Council Licensure Examinations)

17. Find the probability that a student took the exam for the first time, given that the student failed.

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Textbook Question

"In Exercises 5 and 6, use the Fundamental Counting Principle.

5. A student must choose from seven classes to take at 8:00 A.M., four classes to take at 9:00 A.M., and three classes to take at 10:00 A.M. How many ways can the student arrange the schedule?"

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Textbook Question

In Exercises 45-48, use combinations and permutations.

48. An employer must hire 2 people from a list of 13 applicants. In how many ways can the employer choose to hire the two people?

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Textbook Question

In Exercises 49-53, use counting principles to find the probability.

52. A class of 40 students takes a statistics exam. The results are shown in the table at the left. Three students are selected at random. What is the probability that

d. all three students received a B or a C?

66
views
Textbook Question

In Exercises 49-53, use counting principles to find the probability.

52. A class of 40 students takes a statistics exam. The results are shown in the table at the left. Three students are selected at random. What is the probability that

b. all three students received a C or better?

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Textbook Question

In Exercises 49-53, use counting principles to find the probability.

53. A corporation has six male senior executives and four female senior executives. Four senior executives are chosen at random to attend a technology seminar. What is the

probability of choosing

b. four women?

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