91Ó°ÊÓ

News Source Based on data from a Harris Interactive survey, \(40 \%\) of adults say that they prefer to get their news online. Four adults are randomly selected. a. Use the multiplication rule to find the probability that the first three prefer to get their news online and the fourth prefers a different source. That is, find \(P(\mathrm{OOOD})\), where \(\mathrm{O}\) denotes a preference for online news and D denotes a preference for a news source different from online. b. Beginning with \(\mathrm{OOOD}\), make a complete list of the different possible arrangements of those four letters, then find the probability for each entry in the list. c. Based on the preceding results, what is the probability of getting exactly three adults who prefer to get their news online and one adult who prefers a different news source.

Step by step solution

01

Calculate Probability for OOOD (Part a)

To find the probability of the sequence OOOD using the multiplication rule, multiply the probabilities of each independent event. The probability that an adult prefers online news (\text{O}) is 0.4, and the probability that an adult prefers a different source (\text{D}) is 0.6. So, \(P(OOOD) = P(O) * P(O) * P(O) * P(D) = 0.4 * 0.4 * 0.4 * 0.6\).

02

Simplify the Calculation

Calculate the product: \(0.4 * 0.4 = 0.16\), \(0.16 * 0.4 = 0.064\), and finally \(0.064 * 0.6 = 0.0384\). Therefore, \(P(OOOD) = 0.0384\).

03

List All Possible Arrangements (Part b)

The different possible arrangements of the letters OOOD involve permutations of these letters. The total distinct arrangements are: OOOD, OODO, ODOO, DOOO. These are all the possible ways to arrange three 'O's and one 'D'.

04

Calculate Probability for Each Arrangement

Since each arrangement has the same individual probability as step 1, which we found to be 0.0384, we have: \(P(OOOD) = 0.0384\), \(P(OODO) = 0.0384\), \(P(ODOO) = 0.0384\), \(P(DOOO) = 0.0384\).

05

Sum Up Probabilities (Part c)

Sum the probabilities of each unique arrangement to find the probability of exactly three adults preferring online news and one preferring a different source: \(0.0384 + 0.0384 + 0.0384 + 0.0384 = 0.1536\).

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Key Concepts

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

Multiplication Rule

The multiplication rule is a fundamental principle in probability theory. It helps us find the probability of two or more independent events occurring together. For instance, in our exercise, we want to find the probability that the first three out of four selected adults prefer to get their news online, and the fourth prefers a different source. Each event here (each adult’s preference) is independent.

The rule states that the probability of the sequence of events is the product of the probabilities of each individual event. So, for the given problem:

• Probability that an adult prefers online news (\text{O}) is 0.4.
• Probability that an adult prefers a different source (\text{D}) is 0.6.

To find the probability of the specific sequence OOOD, we multiply these probabilities:
\( P(OOOD) = P(O) * P(O) * P(O) * P(D) = 0.4 * 0.4 * 0.4 * 0.6 = 0.0384\).

Permutations

Permutations deal with the arrangement of objects, where the order does matter. In this exercise, we’ll arrange the letters 'OOOD' in all possible ways. Here, we have three 'O's and one 'D', and the permutations of these letters represent different sequences of adults' news preferences.

The total permutations for the letters are:

• OOOD
• OODO
• ODOO
• DOOO

Each arrangement has the same probability computed earlier using the multiplication rule. Since we have four distinct permutations and each has a probability of 0.0384, we will then consider all these arrangements for the final sum.

Independent Events

In probability theory, two events are independent if the occurrence of one event does not affect the occurrence of the other event. This concept is crucial for applying the multiplication rule correctly.

In our problem, each selected adult’s preference for news is an independent event. This means one adult’s preference does not change the probability of another adult’s preference. Hence, multiplying the individual probabilities of preferences gives us the probability of the overall sequence.
So, the event of adult 1 preferring online news and adult 2 also preferring online news are independent events, and the same logic applies to adults 3 and 4. Always ensure events are independent before applying the multiplication rule to avoid errors in probability calculation.

Binomial Probability

The binomial probability formula is useful when dealing with experiments that have two possible outcomes: success or failure. For example, considering 'prefers online news' as success, and 'prefers a different news source' as failure, this scenario fits a binomial distribution.

The binomial probability formula is:\[ P(X = k) = \binom{n}{k} * p^k * (1-p)^{n-k} \]here,

• \(n\) is the number of trials (adults selected = 4)
• \(k\) is the number of successes (adults preferring online news = 3)
• \(p\) is the probability of success on each trial (0.4 for preferring online news)

Using the formula, \( \binom{4}{3} * 0.4^3 * 0.6^1 = 4 * 0.064 * 0.6 = 0.1536 \)Thus, the total probability of exactly 3 out of 4 adults preferring online news and 1 preferring a different source is 0.1536, confirming our earlier calculation by permutation summation.