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Chapter 5: Problem 30

Roll a fair six-sided die. a. What is the probability that the die shows an even number or a number less than 4 on top? b. What is the probability the die shows an odd number or a number greater than 4 on top?

Short Answer

Expert verified
a. The probability that the die shows an even number or a number less than 4 on top is \(\frac{5}{6}\) .\n b. The probability the die shows an odd number or a number greater than 4 on top is \(\frac{2}{3}\)

Step by step solution

01

Identify the Total and Successful Outcomes for First Question

The total outcomes when a six-sided die is rolled are six [1,2,3,4,5,6]. The successful outcomes for the first question are those when the number on the die is even or less than 4. The even numbers are [2,4,6] and the numbers less than 4 are [1,2,3]. After combining and removing duplicates, the successful outcomes are [1,2,3,4,6]. Therefore, the total number of successful outcomes are 5.
02

Calculate the Probability for First Question

Probability is calculated by dividing the number of successful outcomes by the number of total outcomes. \(\frac{Number of successful outcomes}{Total number of outcomes} = \frac{5}{6} \) for part a.
03

Identify the Total and Successful Outcomes for Second Question

The total outcomes are the same, [1,2,3,4,5,6]. The successful outcomes for the second question are those when the number on the die is odd or greater than 4. The odd numbers are [1,3,5] and the numbers greater than 4 are [5,6]. After combining and removing duplicates, the successful outcomes are [1,3,5,6]. Therefore, the total number of successful outcomes are 4.
04

Calculate the Probability for Second Question

Probability is calculated by dividing the number of successful outcomes by the number of total outcomes. \(\frac{Number of successful outcomes}{Total number of outcomes} = \frac{4}{6} \) for part b.

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Key Concepts

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

Dice
A six-sided die is a classic tool used in probability exercises and games around the world. Each side features one of the numbers from 1 to 6. When a die is rolled, each number is equally likely to show up on top, making it a **fair** die. This means that the outcomes 1, 2, 3, 4, 5, and 6 each have a probability of \( \frac{1}{6} \).
When calculating probabilities with dice, it鈥檚 important to identify the total possible outcomes. For instance, a six-sided die has 6 total outcomes, which provide the basis for further calculations of probability in any scenario where the die is used.
Even numbers
Even numbers are those that can be divided by 2 without leaving a remainder. On a six-sided die, the even numbers are:
  • 2
  • 4
  • 6
When you are tasked with determining the probability of rolling an even number with a die, you count these even outcomes. In our specific example, these numbers are 2, 4, and 6.
To find the probability of rolling an even number, divide the number of even outcomes (3) by the total number of possibilities (6): \(\frac{3}{6} = \frac{1}{2} \).This means there's a 50% chance of rolling an even number on a standard six-sided die.
Odd numbers
Odd numbers differ from even numbers in that they have a remainder of 1 when divided by 2. When rolling a six-sided die, the odd numbers you might roll include:
  • 1
  • 3
  • 5
If asked to find the probability of rolling an odd number, you first identify these odd outcomes. In our dice scenario, the numbers 1, 3, and 5 are considered. Thus, the probability calculation involves the 3 odd outcomes out of 6 possible results:\(\frac{3}{6} = \frac{1}{2} \).So, there is also a 50% chance of rolling an odd number with a fair six-sided die.
Successful outcomes
In probability exercises, successful outcomes refer to the outcomes that satisfy the conditions of the problem. To solve probability questions, it is essential to determine which results are classified as "successful."
To solve our initial exercise, let's define successful outcomes based on the given conditions:
  • For part (a): Successful outcomes must be even or less than 4. After eliminating duplicates from combining these criteria, the successful outcomes are [1, 2, 3, 4, 6].
  • For part (b): Successful outcomes must be odd or greater than 4. Again, after combining, the results [1, 3, 5, 6] are counted as successful outcomes.
The probability of an event is calculated as the ratio of successful outcomes to total outcomes. Therefore, the probability for each part is obtained by counting which numbers meet the specified conditions out of the total six possible outcomes of a six-sided die.

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Most popular questions from this chapter

Consider two pairs of grandparents. The first pair has 4 grandchildren, and the second pair has 32 grandchildren. Which of the two pairs is more likely to have between \(40 \%\) and \(60 \%\) boys as grandchildren, assuming that boys and girls are equally likely as children? Why?

a. Fxplain how you could use a random number table to simulate rolling a fair six-sided die 20 times. Assume you wish to find the probability of rolling a I. Then report a line or two of the random number table (or numbers generated by a computer or calculator) and the values that were obtained from it. b. Report the empirical probubility of rolling a 1 from part (u), and compare it with the theoretical probability of rolling a \(1 .\)

Some estimates say that \(10 \%\) of the population is left-handed. We wish to design a simulation to find an empirical probability that if five babies are born on a single day, one or more will be left-handed. Suppose we decide that the even digits \((0,2,4 .\) 6,8 ) will represent left-handed babies and the odd digits will represent right-handed babies. Explain what is wrong with the stated simulation method, and provide a correct method.

A famous study by Amos Tversky and Nobel laureate Daniel Kahneman asked people to consider two hospitals. Hospital \(\mathrm{A}\) is small and has 15 babies born per day. Hospital B has 45 babies born each day. Over one year, cach hospital recorded the number of days that it had more than 609 girls bom. Assuming that \(50 \%\) of all babies are girls, which hospital had the most such days? Or do you think both will have about the same number of days with more than \(60 \%\) girls born? Answer, and explain. (Source: Tversky, Preference, belief, and similarity: Selected Writings, ed. [Cambridge, MA: MIT Press], 205)

Roll a fair six-sided die. a. What is the probability that the die shows an even number or a number greater than 4 on top? b. What is the probability the die shows an odd number or a number less than 3 on top?
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