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Chapter 5: Problem 7

Assume all variables are binomial. (Note: If values are not found in Table B of Appendix \(A,\) use the binomial formula. Belief in UFOs A survey found that \(10 \%\) of Americans believe that they have seen a UFO. For a sample of 10 people, find each probability: a. That at least 2 people believe that they have seen a UFO b. That 2 or 3 people believe that they have seen a UFO c. That exactly 1 person believes that he or she has seen a UFO

Short Answer

Expert verified
a. 0.2639 b. 0.2511 c. 0.3874

Step by step solution

01

Understand the Problem

We are dealing with a binomial distribution problem where the probability of an event (seeing a UFO) occurs with probability \( p = 0.1 \) and \( n = 10 \) trials. Each subproblem asks for different scenarios of success outcomes.
02

Use the Binomial Formula

The binomial probability formula is given by \( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \) where \( \binom{n}{k} \) is the number of combinations of \( n \) items taken \( k \) at a time.
03

Calculate Probability for (a)

We need to find \(P(X \geq 2)\). This can be calculated by finding \(1 - P(X < 2)\), which means \(1 - (P(X = 0) + P(X = 1))\). Calculate each using the formula:\[ P(X = 0) = \binom{10}{0} (0.1)^0 (0.9)^{10} \approx 0.3487 \]\[ P(X = 1) = \binom{10}{1} (0.1)^1 (0.9)^9 \approx 0.3874 \]Now subtract these from 1: \[ P(X \geq 2) = 1 - (0.3487 + 0.3874) = 1 - 0.7361 = 0.2639 \]
04

Calculate Probability for (b)

Calculate the probability for 2 or 3 successes:\[ P(X = 2) = \binom{10}{2} (0.1)^2 (0.9)^8 \approx 0.1937 \]\[ P(X = 3) = \binom{10}{3} (0.1)^3 (0.9)^7 \approx 0.0574 \]Then sum the probabilities: \[ P(2 \leq X \leq 3) = 0.1937 + 0.0574 = 0.2511 \]
05

Calculate Probability for (c)

This is a straightforward calculation using the binomial formula for \( P(X = 1) \), computed earlier:\[ P(X = 1) = 0.3874 \]
06

Summarize Results

Gather the probabilities calculated:a. \( P(X \geq 2) = 0.2639 \)b. \( P(2 \leq X \leq 3) = 0.2511 \)c. \( P(X = 1) = 0.3874 \)

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Key Concepts

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

Probability
Probability is at the core of predicting the likelihood of different events occurring. When we talk about the probability of seeing a UFO in this example, we refer to it as a statistical chance or likelihood that a person might believe they've seen one based on a percentage given鈥攁 10% chance. Here are the basics to help you grasp probability easily:
  • Probability Scale: Runs from 0 to 1, where 0 means an impossible event and 1 means a certain event.
  • Expressing Probability: It's often represented as a fraction, decimal, or percentage.
  • Calculation: To find a probability, you divide the number of ways the event can occur by the total number of possible outcomes.
The trick is to understand that probability gives us a mathematical way to measure uncertainty and enable predictions about the outcomes of various scenarios.
Binomial Formula
The binomial formula is a powerful tool for calculating the probability of a binomial event. A binomial event is one that only has two possible outcomes: success or failure, like flipping a coin or, in this case, a person believing or not believing they've seen a UFO.
  • The formula is: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] Here, \( n \) is the total number of trials, \( k \) is the number of successful trials you want to find the probability for, \( p \) is the probability of a single success, and \( 1-p \) is the probability of a single failure.
  • Interpretation: The formula helps calculate how likely it is for exactly \( k \) successes to occur in \( n \) trials.
  • Computation:To compute \( P(X = k) \), you start by finding \( \binom{n}{k} \), the number of combinations of \( n \) items taken \( k \) at a time.
    • Follow this by raising the probability of success \( p \) to the power of \( k \), and \( (1-p) \) to the power of \( n-k \).
Mastering the binomial formula offers clarity on how repeated trials behave in statistics.
Combinatorics
Combinatorics is all about counting, arranging, and finding the number of possible ways to choose or arrange items. When dealing with binomial probabilities, combinatorics helps determine the number of ways an event can happen.
  • Key Concept: Binomial coefficients, represented as \( \binom{n}{k} \), show how many ways you can choose \( k \) successes in \( n \) trials. It鈥檚 often read as "n choose k."
  • Formula: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \] Where \( n! \) is the factorial of \( n \), meaning you multiply all whole numbers from 1 to \( n \).
This is crucial because it dictates the distribution's shape, impacting how probabilities are calculated and understood. Combinatorics forms the mathematical foundation behind deciding how events combine under certain conditions.
Statistical Analysis
Statistical analysis is the process of collecting, exploring, and presenting large amounts of data to uncover patterns or trends. Using the binomial distribution in statistical analysis helps us understand phenomena where there are two distinct outcomes.
  • Application: By examining things like UFO sightings, statistical analysis provides insights even into uncertain occurrences, helping assess claims or beliefs statistically rather than anecdotally.
  • Evaluate Outcomes: It assists in making informed decisions by predicting and analyzing potential outcomes.
  • Data Interpretation: Analyzing data using binomial distributions can identify trends, variance, and context, making the data actionable.
Whether it鈥檚 market research or understanding public beliefs, like UFO sightings, statistical analysis serves as a guiding light for drawing evidence-based conclusions.

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Most popular questions from this chapter

A survey shows the probability of the number of automobiles that families in a certain housing plan own. Find the mean, variance, and standard deviation for the distribution. $$ \begin{array}{l|ccccc} \boldsymbol{X} & 1 & 2 & 3 & 4 & 5 \\ \hline \boldsymbol{P}(\boldsymbol{X}) & 0.27 & 0.46 & 0.21 & 0.05 & 0.01 \end{array} $$

In a batch of 2000 calculators, there are, on average, 8 defective ones. If a random sample of 150 is selected, find the probability of 5 defective ones.

A roulette wheel has 38 numbers, 1 through \(36,0,\) and \(00 .\) One-half of the numbers from 1 through 36 are red, and the other half are black; 0 and 00 are green. A ball is rolled, and it falls into one of the 38 slots, giving a number and a color. The payoffs (winnings) for a \(\$ 1\) bet are as follows: If a person bets \(\$ 1,\) find the expected value for each. a. Red b. Even c. \(00\) d. Any single number e. \(0\) or \(00\)

Assume all variables are binomial. (Note: If values are not found in Table B of Appendix \(A,\) use the binomial formula. Fiftythree percent of all persons in the U.S. population have at least some college education. Choose 10 persons at random. Find the probability that a. Exactly one-half have some college education b. At least 5 do not have any college education c. Fewer than 5 have some college education

Use the multinomial formula and find the probabilities for each. a. \(n=3, X_{1}=1, X_{2}=1, X_{3}=1, p_{1}=0.5, p_{2}=0.3\) \(\quad p_{3}=0.2\) b. \(n=5, X_{1}=1, X_{2}=3, X_{3}=1, p_{1}=0.7, p_{2}=0.2\) \(\quad p_{3}=0.1\) c. \(n=7, X_{1}=2, X_{2}=3, X_{3}=2, p_{1}=0.4, p_{2}=0.5,\) \(p_{3}=0.1\)
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