The Following 2 Problems Are Binomial Distribution Ones. In Riverside County 54% Of People Suppoort A Proposition. If You Randomly Select 15 People What Is The Probability That At Least 10 Support The Proposition.

Mar 12, 2025 by ADMIN 214 views

The following 2 problems are binomial distribution ones.

Problem 1: Probability of at least 10 people supporting a proposition

In Riverside county, 54% of people support a proposition. If you randomly select 15 people, what is the probability that at least 10 support the proposition?

To solve this problem, we need to use the binomial distribution formula. The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials, where each trial has a constant probability of success.

Let's define the variables:

n = 15 (number of people selected)
p = 0.54 (probability of a person supporting the proposition)
x = 10, 11, 12, 13, 14, 15 (number of people supporting the proposition)

We want to find the probability that at least 10 people support the proposition, which is equivalent to finding the probability that 10 or more people support the proposition.

Using the binomial distribution formula, we can calculate the probability of exactly k people supporting the proposition as:

P(X = k) = (nCk) * (p^k) * ((1-p)^(n-k))

where nCk is the number of combinations of n items taken k at a time.

We can calculate the probability of at least 10 people supporting the proposition by summing the probabilities of 10 or more people supporting the proposition:

P(X ≥ 10) = P(X = 10) + P(X = 11) + P(X = 12) + P(X = 13) + P(X = 14) + P(X = 15)

Using the binomial distribution formula, we can calculate the probabilities as follows:

P(X = 10) = (15C10) * (0.54^10) * (0.46^5) = 0.155 P(X = 11) = (15C11) * (0.54^11) * (0.46^4) = 0.224 P(X = 12) = (15C12) * (0.54^12) * (0.46^3) = 0.224 P(X = 13) = (15C13) * (0.54^13) * (0.46^2) = 0.124 P(X = 14) = (15C14) * (0.54^14) * (0.46^1) = 0.054 P(X = 15) = (15C15) * (0.54^15) * (0.46^0) = 0.018

Summing the probabilities, we get:

P(X ≥ 10) = 0.155 + 0.224 + 0.224 + 0.124 + 0.054 + 0.018 = 0.799

Therefore, the probability that at least 10 people support the proposition is approximately 79.9%.

Problem 2: Probability of at least 10 people supporting a proposition with a given condition

In a survey of 100 people, 60% of people support a proposition. If you randomly select 15 people from the survey, what is the probability that at least 10 people support the proposition, given that the selected people are from the same survey?

To solve this problem, we need to use the binomial distribution formula with the given condition.

Let's define the variables:

n = 15 (number of people selected)
p = 0.6 (probability of a person supporting the proposition in the survey)
x = 10, 11, 12, 13, 14, 15 (number of people supporting the proposition)

We want to find the probability that at least 10 people support the proposition, given that the selected people are from the same survey.

Using the binomial distribution formula, we can calculate the probability of exactly k people supporting the proposition as:

P(X = k) = (nCk) * (p^k) * ((1-p)^(n-k))

where nCk is the number of combinations of n items taken k at a time.

We can calculate the probability of at least 10 people supporting the proposition by summing the probabilities of 10 or more people supporting the proposition:

P(X ≥ 10) = P(X = 10) + P(X = 11) + P(X = 12) + P(X = 13) + P(X = 14) + P(X = 15)

Using the binomial distribution formula, we can calculate the probabilities as follows:

P(X = 10) = (15C10) * (0.6^10) * (0.4^5) = 0.155 P(X = 11) = (15C11) * (0.6^11) * (0.4^4) = 0.224 P(X = 12) = (15C12) * (0.6^12) * (0.4^3) = 0.224 P(X = 13) = (15C13) * (0.6^13) * (0.4^2) = 0.124 P(X = 14) = (15C14) * (0.6^14) * (0.4^1) = 0.054 P(X = 15) = (15C15) * (0.6^15) * (0.4^0) = 0.018

Summing the probabilities, we get:

P(X ≥ 10) = 0.155 + 0.224 + 0.224 + 0.124 + 0.054 + 0.018 = 0.799

Therefore, the probability that at least 10 people support the proposition, given that the selected people are from the same survey, is approximately 79.9%.

Conclusion

In this article, we have discussed two problems related to the binomial distribution. The first problem involved finding the probability that at least 10 people support a proposition in a random sample of 15 people, given that 54% of people in the population support the proposition. The second problem involved finding the probability that at least 10 people support a proposition in a random sample of 15 people, given that 60% of people in a survey support the proposition.

We have used the binomial distribution formula to solve these problems and have found that the probability that at least 10 people support the proposition is approximately 79.9% in both cases.

The binomial distribution is a powerful tool for modeling the number of successes in a fixed number of independent trials, and it has many applications in statistics and probability theory. By understanding the binomial distribution, we can better analyze and interpret data in a wide range of fields, from medicine and social sciences to business and economics.

Problem 1: Probability of at least 10 people supporting a proposition

In Riverside county, 54% of people support a proposition. If you randomly select 15 people, what is the probability that at least 10 support the proposition?

Let's define the variables:

n = 15 (number of people selected)
p = 0.54 (probability of a person supporting the proposition)
x = 10, 11, 12, 13, 14, 15 (number of people supporting the proposition)

We want to find the probability that at least 10 people support the proposition, which is equivalent to finding the probability that 10 or more people support the proposition.

Using the binomial distribution formula, we can calculate the probability of exactly k people supporting the proposition as:

P(X = k) = (nCk) * (p^k) * ((1-p)^(n-k))

where nCk is the number of combinations of n items taken k at a time.

We can calculate the probability of at least 10 people supporting the proposition by summing the probabilities of 10 or more people supporting the proposition:

P(X ≥ 10) = P(X = 10) + P(X = 11) + P(X = 12) + P(X = 13) + P(X = 14) + P(X = 15)

Using the binomial distribution formula, we can calculate the probabilities as follows:

Summing the probabilities, we get:

P(X ≥ 10) = 0.155 + 0.224 + 0.224 + 0.124 + 0.054 + 0.018 = 0.799

Therefore, the probability that at least 10 people support the proposition is approximately 79.9%.

Problem 2: Probability of at least 10 people supporting a proposition with a given condition

To solve this problem, we need to use the binomial distribution formula with the given condition.

Let's define the variables:

n = 15 (number of people selected)
p = 0.6 (probability of a person supporting the proposition in the survey)
x = 10, 11, 12, 13, 14, 15 (number of people supporting the proposition)

We want to find the probability that at least 10 people support the proposition, given that the selected people are from the same survey.

Using the binomial distribution formula, we can calculate the probability of exactly k people supporting the proposition as:

P(X = k) = (nCk) * (p^k) * ((1-p)^(n-k))

where nCk is the number of combinations of n items taken k at a time.

We can calculate the probability of at least 10 people supporting the proposition by summing the probabilities of 10 or more people supporting the proposition:

P(X ≥ 10) = P(X = 10) + P(X = 11) + P(X = 12) + P(X = 13) + P(X = 14) + P(X = 15)

Using the binomial distribution formula, we can calculate the probabilities as follows:

Summing the probabilities, we get:

P(X ≥ 10) = 0.155 + 0.224 + 0.224 + 0.124 + 0.054 + 0.018 = 0.799

Therefore, the probability that at least 10 people support the proposition, given that the selected people are from the same survey, is approximately 79.9%.

Q&A

Q: What is the binomial distribution?

Q: How do I calculate the probability of at least 10 people supporting a proposition?

A: To calculate the probability of at least 10 people supporting a proposition, you need to use the binomial distribution formula. You can calculate the probability of exactly k people supporting the proposition as:

P(X = k) = (nCk) * (p^k) * ((1-p)^(n-k))

where nCk is the number of combinations of n items taken k at a time.

Q: What is the difference between the binomial distribution and the normal distribution?

A: The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials, where each trial has a constant probability of success. The normal distribution, on the other hand, is a continuous probability distribution that models the distribution of a continuous random variable.

Q: Can I use the binomial distribution to model the number of people supporting a proposition in a survey?

A: Yes, you can use the binomial distribution to model the number of people supporting a proposition in a survey. However, you need to make sure that the survey is representative of the population and that the sample size is large enough to provide reliable results.

Q: How do I calculate the probability of at least 10 people supporting a proposition with a given condition?

A: To calculate the probability of at least 10 people supporting a proposition with a given condition, you need to use the binomial distribution formula with the given condition. You can calculate the probability of exactly k people supporting the proposition as:

P(X = k) = (nCk) * (p^k) * ((1-p)^(n-k))

where nCk is the number of combinations of n items taken k at a time.

Q: What is the difference between the binomial distribution and the Poisson distribution?

A: The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials, where each trial has a constant probability of success. The Poisson distribution, on the other hand, is a discrete probability distribution that models the number of events occurring in a fixed interval of time or space.

Q: Can I use the binomial distribution to model the number of people supporting a proposition in a social media campaign?

A: Yes, you can use the binomial distribution to model the number of people supporting a proposition in a social media campaign. However, you need to make sure that the social media campaign is representative of the population and that the sample size is large enough to provide reliable results.

Q: How do I interpret the results of a binomial distribution analysis?

A: To interpret the results of a binomial distribution analysis, you need to understand the meaning of the probability values. A probability value of 0.5 or higher indicates that the event is likely to occur, while a probability value of 0.5 or lower indicates that the event is unlikely to occur.

Q: Can I use the binomial distribution to model the number of people supporting a proposition in a political election?

A: Yes, you can use the binomial distribution to model the number of people supporting a proposition in a political election. However, you need to make sure that the election is representative of the population and that the sample size is large enough to provide reliable results.