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Chapter 5: Problem 21

Assume that when adults with smartphones are randomly selected, \(54 \%\) use them in meetings or classes (based on data from an LG Smartphone survey). If 8 adult smartphone users are randomly selected, find the probability that exactly 6 of them use their smartphones in meetings or classes.

Short Answer

Expert verified
The probability is approximately 0.276.

Step by step solution

01

Define the parameters

Identify the key parameters in the problem. Let the total number of adults be denoted by n and the probability of one adult using a smartphone in meetings or classes be denoted by p. Here, n = 8 and p = 0.54.
02

Use the Binomial Probability Formula

To find the probability of exactly 6 adults using smartphones in meetings or classes, the binomial probability formula is used. The formula is given by:\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]where \ n = 8, \ k = 6, \ p = 0.54, and \ 1 - p = 0.46.
03

Substitute the values

Substitute the values into the binomial probability formula:\[ P(X = 6) = \binom{8}{6} (0.54)^6 (0.46)^2 \]
04

Calculate the binomial coefficient

Calculate the binomial coefficient \( \binom{8}{6} \), which represents the number of ways to choose 6 successes out of 8 trials:\[ \binom{8}{6} = \frac{8!}{6!(8-6)!} = \frac{8!}{6!2!} = 28 \]
05

Compute the probability

Now, compute the probability by multiplying the binomial coefficient by the probabilities:\[ P(X = 6) = 28 (0.54)^6 (0.46)^2 \]
06

Perform the calculations

Calculate the following values:\[ (0.54)^6 鈮� 0.0467 \]\[ (0.46)^2 鈮� 0.2116 \]Multiplying these values gives:\[ P(X = 6) 鈮� 28 \times 0.0467 \times 0.2116 \]\[ P(X = 6) 鈮� 0.276 \]

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Key Concepts

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

binomial distribution
A binomial distribution is a type of probability distribution that summarizes the likelihood that a value will take one of two independent values under a given set of parameters or assumptions. It is used to model the number of successes in a fixed number of trials in an experiment. In this problem, our 'experiment' is selecting 8 adults with smartphones, and our two possible outcomes are using a smartphone in meetings or classes (success) and not using a smartphone in meetings or classes (failure). The probability that one adult uses their smartphone in meetings or classes is 0.54, and the experiment is repeated 8 times (one for each adult).
statistical probability
Statistical probability refers to the chance or likelihood of an event occurring based on data or previous experiences. In our case, it is derived from the survey conducted by LG Smartphones, which indicates that 54% of adults use their smartphones in meetings or classes. We use this probability to predict the number of adults out of 8 who will likely use their smartphones. The concept helps in understanding how often an outcome will occur, based on empirical evidence.
probability formula
Probability formulas are essential for calculating the likelihood of various outcomes. In binomial distribution, the formula to find the probability of obtaining exactly k successes in n trials is given by:
\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] \ where: \
    \
  • n is the total number of trials (in our example, n=8)
    • \
  • k is the number of successes we are interested in (in our example, k=6)
    • \
  • p is the probability of success in a single trial (here, p=0.54)
    • \
  • (1-p) is the probability of failure (here, 0.46)
    • \
\This formula combines the binomial coefficient with the probabilities of success and failure raised to the appropriate powers.
binomial coefficient
The binomial coefficient, also known as 'n choose k', is a coefficient that appears in the binomial theorem. It is denoted by \( \binom{n}{k} \), and it represents the number of ways to choose k successes out of n trials. The binomial coefficient is calculated using the formula:
\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \] \ where: \
    \
  • n! (n factorial) is the product of all positive integers less than or equal to n
    • \
  • k! is the product of all positive integers less than or equal to k
    • \
  • (n-k)! is the product of all positive integers less than or equal to (n-k)
    • \
\In our example, we need \( \binom{8}{6} \), calculated as follows: \
\[ \binom{8}{6} = \frac{8!}{6!(8-6)!} = \frac{8!}{6!2!} = 28 \] \
This value tells us there are 28 ways to choose 6 users out of 8 who use their smartphones in meetings or classes.

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Most popular questions from this chapter

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