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Chapter 12: Problem 10

Languages in Canada. Canada has two official languages: English and French. Choose a Canadian at random and ask, "What is your mother tongue?" Here is the distribution of responses, combining many separate languages: \(\underline{5}\) \begin{tabular}{|l|c|c|c|} \hline Language & English & French & Other \\ \hline Probability & \(0.57\) & \(0.21\) & \(?\) \\ \hline \end{tabular} a. What is the probability that a Canadian's mother tongue is either English or French? b. What probability should replace "?" in the distribution? c. What is the probability that a Canadian's mother tongue is not English?

Short Answer

Expert verified
a. 0.78 b. 0.22 c. 0.43

Step by step solution

01

Add Known Probabilities

The problem provides the probabilities of the mother tongues being English and French: 0.57 and 0.21 respectively. To find the probability that a Canadian's mother tongue is either English or French, we simply add these two probabilities together: \( 0.57 + 0.21 \).
02

Calculate Probability for Other Languages

Since the total probability for all possible outcomes must equal 1, the probability of 'Other' languages can be found by subtracting the combined probability of English and French from 1. Thus, \( 1 - (0.57 + 0.21) \).
03

Find Probability of Not English

The probability that a Canadian's mother tongue is not English is the sum of the probabilities for French and Other. Using the complement of English, the calculation is \( 1 - 0.57 \).

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Key Concepts

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

Complement Rule
The complement rule is a fundamental concept in probability. It helps us find the probability of an event not happening. If we have the probability of an event happening, its complement is the event not happening.

To use the complement rule, remember:
  • The sum of the probabilities of an event and its complement is always equal to 1.
  • If the probability of an event "A" is known, then the complement, or the probability of not "A", is calculated as: \[ P(\text{not } A) = 1 - P(A) \]

For example, if the probability that a Canadian's mother tongue is English is 0.57, then using the complement rule, the probability that it is not English is:
  • \[ P(\text{not English}) = 1 - 0.57 = 0.43 \]
This tells us there's a 43% chance a Canadian does not have English as their mother tongue.
Addition Rule in Probability
The addition rule is used to find the probability of either of two mutually exclusive events happening. When events cannot happen simultaneously, they are mutually exclusive.

The rule states:
  • If "A" and "B" are mutually exclusive events, then the probability of either "A" or "B" occurring is simply the sum of their individual probabilities:
  • \[ P(A \text{ or } B) = P(A) + P(B) \]

Applying this to our problem finds the probability of a Canadian's mother tongue being either English or French:
  • Given that: \[P(\text{English}) = 0.57 \] and \[ P(\text{French}) = 0.21 \]
  • The probability of English or French = \[ 0.57 + 0.21 = 0.78 \]
So, there is a 78% chance that a Canadian's mother tongue is either English or French.
Sum of Probabilities
The sum of all probabilities in any probability distribution must equal 1. This ensures that one of the possible outcomes will occur. If the total doesn't equal 1, the distribution is not complete or correct.

In our example, we need to determine the missing probability for the "Other" languages:

To do this:
  • Add the known probabilities for English and French, which are \(0.57 + 0.21 = 0.78\).
  • Subtract this sum from 1 (the total probability of all possible outcomes). Thus,\[ P(\text{Other}) = 1 - 0.78 = 0.22 \]
Thus, the probability that a Canadian's mother tongue is in the "Other" category is 0.22, ensuring that the sum of all probabilities is indeed equal to 1.

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Most popular questions from this chapter

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Unusual Dice. Nonstandard dice can produce interesting distributions of outcomes. You have two balanced, six-sided dice. One is a standard die, with faces having \(1,2,3,4,5\), and 6 spots. The other die has three faces with 0 spots and three faces with 6 spots. Find the probability distribution for the total number of spots \(Y\) on the up-faces when you roll these two dice. Hint: Start with a picture like Figure \(12.2\) (page 276 ) for the possible up-faces. Label the three 0 faces on the second die \(0 \mathrm{a}, 0 \mathrm{~b}, 0 \mathrm{c}\) in your picture and similarly distinguish the three 6 faces.)

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