Chapter 16: Problem 14

Two dice are thrown. Use the sample space (2.4) to answer the following questions. (a) What is the probability of being able to form a two-digit number greater than 33 with the two numbers on the dice? (Note that the sample point 1, 4 yields the two-digit number 41 which is greater than 33, etc.) (b) Repeat part (a) for the probability of being able to form a two-digit number greater than or equal to 42 (c) Can you find a two-digit number (or numbers) such that the probability of being able to form a larger number is the same as the probability of being able to form a smaller number? [See note, part (a).]

Short Answer

Expert verified

(a) 1/2, (b) 5/12, (c) The number is 36.

Step by step solution

Title - Understanding the Sample Space

When two dice are thrown, each die has 6 faces. Therefore, the total number of possible outcomes is 6 × 6 = 36. The outcomes can be represented as pairs (x, y) where x and y vary from 1 to 6. The sample space S = {(1,1), (1,2), …, (6,6)}.

Title - Identifying Two-Digit Numbers Greater than 33 (Part a)

To form a two-digit number greater than 33, consider x as the tens digit and y as the units digit. We need to check pairs where 10x + y > 33. Find and count all pairs satisfying 10x + y > 33.

Title - Probability Calculation for Part (a)

Identify valid pairs: (4,1), (4,2), ..., (4,6); (5,1), (5,2), ..., (5,6); (6,1), (6,2), ..., (6,6). Total such outcomes = 18. Probability P = Number of favorable outcomes / Total possible outcomes = 18 / 36 = 1/2.

Title - Identifying Two-Digit Numbers Greater than or Equal to 42 (Part b)

For numbers greater than or equal to 42, check pairs where 10x + y >= 42. Identify these pairs.

Title - Probability Calculation for Part (b)

Valid pairs are: (4,2), (4,3), ..., (4,6); (5,1), (5,2), ..., (5,6); (6,1), (6,2), ..., (6,6). Total such outcomes = 15. Probability P = 15 / 36 = 5/12.

Title - Finding Two-Digit Number with Equal Probabilities (Part c)

Want P(10x + y > k) = P(10x + y < k). Both probabilities should be equal for some k. Calculate probabilities stepwise and compare.

Title - Verification of Equal Probabilities (Part c)

Finding k = 36 satisfies P(10x + y > 36) = P(10x + y < 36) = 18/36 = 1/2. Therefore, 36 is the number that splits equal probability for larger and smaller numbers.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Key Concepts

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

Sample Space

When discussing probability, especially with dice, it’s important to understand the concept of the sample space. The sample space is a set of all possible outcomes. For two dice, each die has 6 faces. When thrown together, each face on the first die can pair with each face on the second, giving us a total of 36 possible outcomes. These pairs can be represented as (x, y), where both x and y can be any number from 1 to 6. So, for two thrown dice, the sample space S consists of pairs like (1,1), (1,2), all the way to (6,6). Identifying all these possibilities is the foundation of calculating probabilities.

Favorable Outcomes

Favorable outcomes are the set of outcomes that satisfy our event of interest. For example, in exercise (a), we need to find outcomes where we can form a two-digit number greater than 33 using the numbers rolled on two dice. To build a two-digit number, consider the first number (x) as the tens place and the second number (y) as the units place. We need pairs where the number 10x + y is greater than 33. By listing all valid combinations like (4,1), (4,2), ..., (6,6), we find that there are 18 favorable outcomes. It’s these specific pairs that we use to determine the probability for this particular scenario.

Two-Digit Numbers

Forming two-digit numbers with dice involves creative thinking. In exercise (a), a two-digit number, more than 33, needs to be structured as 10 times the first roll plus the second roll. This brings in a compelling twist as we consider our pairs like (4,1) which gives 41, and (6,5) which gives 65. Moving on to exercise (b), we need pairs such that the two-digit number is at least 42. Combinations like (4,2) producing 42 start to become valid now. Hence, the new numbers stretch from 42 all the way to 66. This detailed breakdown helps understand why certain numbers qualify and how probabilities change as criteria change.

Probability Calculation

Once we have our sample space and know our favorable outcomes, we can easily calculate probabilities. Probability itself is the ratio of the number of favorable outcomes to the total number of possible outcomes. For example, in exercise (a), we have 18 favorable two-digit numbers greater than 33 out of a total of 36 possible outcomes. So, the probability P = 18 / 36 = 1/2. Similarly, for exercise (b), the probability of forming a number greater than or equal to 42 is found using 15 favorable outcomes over 36, giving us P = 15 / 36 = 5/12. Understanding this division and simplification helps in grasping the core of probability calculation.