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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

.Land in Canada. Canada's national statist ics agency, Statistics Canada, says that the land area of Canada is \(9,094,000\) square kilometers. Of this land, 4,176, 000 square kilometers are forested. Choose a square kilometer of land in Canada at random. (Assume a selected square is classified as either forested or not forested.) a. What is the probability that the area you chose is forested? b. What is the probability that it is not forested?

Playing Pick 4. The Pick 4 games in many state lotteries announce a four-digit winning number each day. Each of the 10,000 possible numbers 0000 to 9999 has the same chance of winning. You win if your choice matches the winning digits. Suppose your chosen number is \(5974 .\) a. What is the probability that the winning number matches your number exactly? b. What is the probability that the winning number has the same digits as your number in any order?

Random Numbers. Many random number generators allow users to specify the range of the random numbers to be produced. Suppose you specify that the random number \(Y\) can take any value between 0 and 2 . Then the density curve of the outcomes has constant height between 0 and 2 and height 0 elsewhere. a. Is the random variable \(Y\) discrete or continuous? Why? b. What is the height of the density curve between 0 and 2? Draw a graph of the density curve. c. Use your graph from part (b) and the fact that probability is area under the curve to find \(P(Y \leq 1)\).

Probability Models? In each of the following situations, state whether or not the given assignment of probabilities to individual outcomes is legitimate- that is, satisfies the rules of probability. Remember, a legitimate model need not be a practically reasonable model. If the assignment of probabilities is not legitimate, give specific reasons for your answer. a. Roll a six-sided die and record the count of spots on the upface: $$ \begin{array}{lll} P(1)=0 & P(2)=1 / 6 & P(3)=1 / 3 \\ P(4)=1 / 3 & P(5)=1 / 6 & P(6)=0 \end{array} $$ b. Deal a card from a shuffled deck: $$ \begin{array}{rlrl} P(\text { clubs }) & =12 / 52 & P(\text { diamonds }) & =12 / 52 \\ P(\text { hearts }) & =12 / 52 & P(\text { spades }) & =16 / 52 \end{array} $$ c. Choose a college student at random and record sex and enrollment status: $$ \begin{array}{rlrl} P(\text { female full-time }) & =0.56 & P(\text { male full-time }) & =0.44 \\\ P(\text { female part-time }) & =0.24 & P(\text { male part-time }) & =0.17 \end{array} $$

Sample Space. In each of the following situations, describe a sample space \(S\) for the random phenomenon. a. A basket ball player shoots four free throws. You record the sequence of hits and misses. b. A basket ball player shoots four free throws. You record the number of baskets she makes.
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