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Chapter 12: Problem 29

In a table of random digits such as Table B, each digit is equally likely to be any of \(0,1,2,3,4,5,6,7,8\), or 9 . What is the probability that a digit in the table is 7 or greater? a. \(7 / 10\) b. \(4 / 10\) c. \(3 / 10\)

Short Answer

Expert verified
The probability is \( \frac{3}{10} \) (option c).

Step by step solution

01

Identify relevant digits

To find the probability of a digit being 7 or greater, first identify the digits that meet this condition. These digits are: 7, 8, and 9.
02

Count favorable outcomes

Count the number of favorable digits. Since 7, 8, and 9 are the digits from 7 to 9, there are 3 favorable outcomes in this situation.
03

Count all possible outcomes

Identify the total number of possible outcomes, which are all the digits from 0 to 9. This gives a total of 10 possible digits.
04

Calculate probability

The probability of an event is given by the ratio of favorable outcomes to the total outcomes. So, the probability that a digit is 7 or greater is \( \frac{3}{10} \).

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Key Concepts

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

Random Digits
Random digits are numbers that can occur in any position within a sequence, without any discernible pattern. Think of them as a "lottery" where each digit from 0 to 9 has an equal chance of being selected. In a table of random digits, every number has the same probability of appearing. This randomness is crucial for many statistical and probability calculations because it ensures unbiased outcomes. Imagine flipping a coin with ten sides, each marked with a different digit from 0 to 9. Every time you flip it, each side has the same likelihood of showing up.
Equally Likely Outcomes
Equally likely outcomes refer to scenarios where each possible result has the same chance of happening. In the context of random digits, this means that any digit from 0 to 9 has an equal probability of being chosen. No number is more or less likely than the others. This forms the basis for, and simplifies, probability calculations, because it ensures a uniform distribution of all potential outcomes. For example, if you have ten marbles, each with a different number from 0 to 9, and you draw one at random, there is an equal chance for any marble, making them equally likely outcomes.
Favorable Outcomes
Favorable outcomes are the specific results that we are interested in out of all possible outcomes. When calculating probability, these are the outcomes that "favor" the event we're considering. In the exercise, the favorable outcomes are digits that are 7 or greater. This is because we want to find the probability of drawing such a digit from a set of random digits. So, we look at the numbers 7, 8, and 9 and count them as favorable. If you have a bag of random digits and only need the numbers equal to or above 7, then only those figures are favorable.
Event Probability Calculation
Calculating an event's probability involves a simple ratio of favorable outcomes to total possible outcomes. In mathematical terms, it is represented by the formula \( P( ext{Event}) = \frac{ ext{Number of Favorable Outcomes}}{ ext{Total Number of Outcomes}} \). For the exercise, 7, 8, and 9 are the favorable outcomes, totaling 3. All possible digits range from 0 to 9, giving us 10 possible outcomes. Therefore, the probability of picking a digit that is 7 or greater is given by \( \frac{3}{10} \). This fraction tells us that out of every 10 random digits chosen, about 3 will likely be 7 or greater.

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Most popular questions from this chapter

Probability Says ... Probability is a measure of how likely an event is to occur. Match one of the probabilities that follow with each statement of likelihood given. (The probability is usually a more exact measure of likelihood than is the verbal statement.) \(\begin{array}{lllllll}0 & 0.05 & 0.45 & 0.50 & 0.55 & 0.95 \quad 1\end{array}\) a. This event is impossible. It can never occur. b. This event is just as likely to occur as it is to not occur. c. This event is very likely, but it will not occur once in a while in a long sequence of trials. d. This event will occur slightly less often than not.

A Taste Test. A tea-drinking Canadian friend of yours claims to have a very refined palate. She tells you that she can tell if, in preparing a cup of tea, milk is first added to the cup and then hot tea poured into the cup or the hot tea is first poured into the cup and then the milk is added. \(1 .\) To test her claims, you prepare six cups of tea. Three have the milk added first and the other three the tea first. In a blind taste test, your friend tastes all six cups and is asked to identify the three that had the milk added first. a. How many different ways are there to select three of the six cups? (Hint: See Example 12.8, page 281.) b. If your friend is just guessing, what is the probability that she correctly identifies the three cups with the milk added first?
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