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Chapter 6: Problem 3

Spell-checking Spell-checking software catches "nonword errors," which result in a string of letters that is not a word, as when "the" is typed as "teh." When undergraduates are asked to write a 250 -word essay (without spell- checking), the number \(X\) of nonword errors has the following distribution: $$ \begin{array}{lccccc} \hline \text { Value: } & 0 & 1 & 2 & 3 & 4 \\ \text { Probability: } & 0.1 & 0.2 & 0.3 & 0.3 & 0.1 \\ \hline \end{array} $$ (a) Write the event "at least one nonword error" in terms of \(X .\) What is the probability of this event? (b) Describe the event \(X \leq 2\) in words. What is its probability? What is the probability that \(X<2 ?\)

Short Answer

Expert verified
(a) \(P(X \geq 1) = 0.9\). (b) \(X \leq 2\) is 'zero, one, or two errors'; \(P(X \leq 2) = 0.6\); \(P(X < 2) = 0.3\).

Step by step solution

01

Define 'At Least One Error'

The phrase "at least one nonword error" refers to the event that there is one or more nonword errors in the essay. Mathematically, this means any number other than zero. Therefore, the event can be written as \(X \geq 1\) or \(X = 1, 2, 3,\) or \(4\).
02

Calculate Probability for 'At Least One Error'

To calculate the probability of having at least one error, let's first consider the probability of the complementary event, which is having zero errors \(X = 0\). From the distribution: \(P(X = 0) = 0.1\). Therefore, the probability of having at least one error is \(1 - P(X = 0) = 1 - 0.1 = 0.9\).
03

Describe Event \(X \leq 2\) in Words

The event \(X \leq 2\) means the number of nonword errors is two or less. In words, this means 'the essay contains zero, one, or two nonword errors.'
04

Calculate Probability for \(X \leq 2\)

To find the probability of \(X \leq 2\), sum the probabilities of 0, 1, and 2 nonword errors: \(P(X = 0) + P(X = 1) + P(X = 2) = 0.1 + 0.2 + 0.3 = 0.6\).
05

Calculate Probability for \(X < 2\)

The event \(X < 2\) means the number of errors is less than two, i.e., zero or one error. The probability is found by adding the probabilities of 0 and 1 errors: \(P(X = 0) + P(X = 1) = 0.1 + 0.2 = 0.3\).

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Key Concepts

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

Nonword Errors
Nonword errors are simple mistakes that occur when a sequence of letters does not form a valid word. It's a common issue many face when typing quickly, like typing "teh" instead of "the." In a spell-checking exercise where students write a 250-word essay, nonword errors are tracked to evaluate their impact on writing quality. The number of such errors is represented by the random variable \( X \), which ranges from 0 (no errors) to 4 (more noticeable errors). Each value corresponds to a certain probability, showing how likely each scenario is when students attempt the exercise without spell-check assistance. These errors highlight common typing issues that reflect in real-world writing tasks.
Probability Calculation
Calculating probabilities helps to understand the likelihood of various outcomes. The distribution of nonword errors shows:
  • 0 errors with a 10% probability
  • 1 error with a 20% probability
  • 2 errors with a 30% probability
  • 3 errors with a 30% probability
  • 4 errors with a 10% probability
One example is calculating the probability of having at least one nonword error. The complement rule is handy here. First, identify the probability of having zero errors, which is 0.1. Then, subtract this from 1 to find the reverse scenario, so the calculation is: \( 1 - 0.1 = 0.9 \). This means a 90% chance that the essay includes at least one mistake.
Statistical Events
Statistical events describe possible outcomes that can be assessed using probabilities. Events like "at least one nonword error" or "two or fewer errors" are analyzed using their respective values in the probability distribution. Consider the event \( X \leq 2 \), meaning there are two or fewer errors.
To find this probability, simply add the probabilities of the individual events where \( X = 0 \), \( X = 1 \), and \( X = 2 \). Mathematically, it's represented as: \[P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2) = 0.1 + 0.2 + 0.3 = 0.6\] This sums up to a 60% likelihood of two or fewer nonword errors occurring. Another event, \( X < 2 \), only includes values \( X = 0 \) or \( X = 1 \), giving a probability of:\[P(X < 2) = P(X = 0) + P(X = 1) = 0.1 + 0.2 = 0.3\] Thus, a 30% chance exists for less than two errors to occur, making it crucial to know how often certain writing mistakes happen.

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Most popular questions from this chapter

Marti decides to keep placing a \(\$ 1\) bet on number 15 in consecutive spins of a roulette wheel until she wins. On any spin, there's a 1 -in- 38 chance that the ball will land in the 15 slot. (a) How many spins do you expect it to take until Marti wins? Justify your answer. (b) Would you be surprised if Marti won in 3 or fewer spins? Compute an appropriate probability to support your answer.

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