If you happen to be viewing the article According to a recent survey, 31 percent of the residents of a certain state who are age 25 years or older have a bachelor’s degree. A random sample of 50 residents of the state, age 25 years or older, will be selected. Let the random variable B represent the number in the sample who have a bachelor’s degree. What is the probability that B will equal 40 ? ? on the website Math Hello Kitty, there are a couple of convenient ways for you to navigate through the content. You have the option to simply scroll down and leisurely read each section at your own pace. Alternatively, if you’re in a rush or looking for specific information, you can swiftly click on the table of contents provided. This will instantly direct you to the exact section that contains the information you need most urgently.
Check out education in a state – 31% of people 25 or older have a bachelor’s degree. Now, think about picking 50 people at random. What’s the chance that exactly 40 of them have a bachelor’s degree?
According to a recent survey, 31 percent of the residents of a certain state who are age 25 years or older have a bachelor’s degree. A random sample of 50 residents of the state, age 25 years or older, will be selected. Let the random variable B represent the number in the sample who have a bachelor’s degree. What is the probability that B will equal 40 ?
The probability of having exactly 40 residents with a bachelor’s degree in a random sample of 50 from this state is incredibly low. Using the binomial probability formula and the provided parameters:
n (sample size) = 50 k (successes, residents with a bachelor’s degree) = 40
p (probability of success, resident having a bachelor’s degree) = 0.31
The probability, P(B = 40), can be calculated as:
P(B = 40) = (50 choose 40) * (0.31)^40 * (0.69)^10
Calculating each component, we get:
(50 choose 40) = approximately 211,876,000 (0.31)^40 = approximately 0.00000000000732 (0.69)^10 = approximately 0.29644696
Therefore, P(B = 40) = 211,876,000 * 0.00000000000732 * 0.29644696 ≈ 1.134 x 10^-12.
This means that the probability of finding exactly 40 residents with a bachelor’s degree in a random sample of 50 is approximately 1.134 times 10 raised to the power of -12. This is an extremely small probability, suggesting that it’s very unlikely to observe this specific outcome.
In simpler terms, the chance of observing this exact scenario is incredibly rare, like drawing four identical winning lotto numbers in a row. While possible, it’s highly improbable.
What is Probability in the context of Mathematics?
In mathematics, probability is a branch of study that deals with the likelihood or chance of different outcomes occurring in a given event or experiment. It provides a way to quantify uncertainty and randomness. The basic idea is to assign a numerical measure between 0 and 1 to the likelihood of an event, where 0 indicates impossibility and 1 indicates certainty.
The probability of an event A is denoted by P(A), and it satisfies the following properties:
- Probability is always between 0 and 1: 0 ≤ P(A) ≤ 1.
- The probability of the entire sample space (all possible outcomes) is 1: P(S) = 1, where S is the sample space.
- If A and B are mutually exclusive events (they cannot occur simultaneously), then the probability of either A or B occurring is the sum of their individual probabilities: P(A or B) = P(A) + P(B).
- The probability of the complement of an event A (denoted as A’) is 1 minus the probability of A: P(A’) = 1 – P(A).
Probability theory is widely used in various fields, including statistics, economics, physics, biology, and computer science, to model and analyze uncertain or random phenomena. It provides a foundation for making predictions, decisions, and inferences based on incomplete information.
Thank you so much for taking the time to read the article titled According to a recent survey, 31 percent of the residents of a certain state who are age 25 years or older have a bachelor’s degree. A random sample of 50 residents of the state, age 25 years or older, will be selected. Let the random variable B represent the number in the sample who have a bachelor’s degree. What is the probability that B will equal 40 ? written by Math Hello Kitty. Your support means a lot to us! We are glad that you found this article useful. If you have any feedback or thoughts, we would love to hear from you. Don’t forget to leave a comment and review on our website to help introduce it to others. Once again, we sincerely appreciate your support and thank you for being a valued reader!
Source: Math Hello Kitty
Categories: Math
