Sam's Closet Contains Blue And Green Shirts. He Has Eight Blue Shirts And Seven Green Shirts. Five Of The Blue Shirts Have Stripes, And Four Of The Green Shirts Have Stripes. What Is The Probability That Sam Randomly Chooses A Shirt That Is Blue Or Has
Introduction
Probability is a fundamental concept in mathematics that helps us understand the likelihood of events occurring. In real-life scenarios, probability is used to make informed decisions and predictions. In this article, we will explore a scenario where Sam has a closet containing blue and green shirts, and we need to find the probability of randomly choosing a shirt that is blue or has stripes.
The Scenario
Sam's closet contains blue and green shirts. He has eight blue shirts and seven green shirts. Five of the blue shirts have stripes, and four of the green shirts have stripes. We need to find the probability of randomly choosing a shirt that is blue or has stripes.
Defining the Sample Space
To find the probability, we need to define the sample space, which is the set of all possible outcomes. In this case, the sample space consists of all the shirts in Sam's closet. We can represent the sample space as a set of blue shirts (B) and green shirts (G), with each shirt having a stripe or not.
Calculating the Probability
To calculate the probability, we need to find the number of favorable outcomes (blue shirts or shirts with stripes) and divide it by the total number of possible outcomes (all shirts in the sample space).
Let's break down the calculation:
- Number of blue shirts: 8
- Number of green shirts: 7
- Number of blue shirts with stripes: 5
- Number of green shirts with stripes: 4
We can find the total number of blue shirts or shirts with stripes by adding the number of blue shirts with stripes and the number of green shirts with stripes:
- Total number of blue shirts or shirts with stripes: 5 (blue shirts with stripes) + 4 (green shirts with stripes) = 9
However, we also need to consider the blue shirts without stripes. Since there are 8 blue shirts in total, and 5 of them have stripes, the number of blue shirts without stripes is:
- Number of blue shirts without stripes: 8 (total blue shirts) - 5 (blue shirts with stripes) = 3
Now, we can find the total number of blue shirts or shirts with stripes by adding the number of blue shirts with stripes, the number of green shirts with stripes, and the number of blue shirts without stripes:
- Total number of blue shirts or shirts with stripes: 5 (blue shirts with stripes) + 4 (green shirts with stripes) + 3 (blue shirts without stripes) = 12
However, this is incorrect because we are counting the blue shirts without stripes twice. We should only count the blue shirts with stripes and the green shirts with stripes once.
The correct total number of blue shirts or shirts with stripes is:
- Total number of blue shirts or shirts with stripes: 5 (blue shirts with stripes) + 4 (green shirts with stripes) = 9
Now, we can find the probability of randomly choosing a shirt that is blue or has stripes by dividing the total number of blue shirts or shirts with stripes by the total number of shirts in the sample space:
- Probability: 9 (total number of blue shirts or shirts with stripes) / 15 (total number of shirts in the sample space) = 3/5
Conclusion
In this article, we explored a scenario where Sam has a closet containing blue and green shirts, and we needed to find the probability of randomly choosing a shirt that is blue or has stripes. We defined the sample space, calculated the number of favorable outcomes, and divided it by the total number of possible outcomes to find the probability. The probability of randomly choosing a shirt that is blue or has stripes is 3/5.
Real-Life Applications
Probability is used in various real-life scenarios, such as:
- Insurance: Insurance companies use probability to determine the likelihood of an event occurring and set premiums accordingly.
- Finance: Financial institutions use probability to make investment decisions and manage risk.
- Medicine: Medical professionals use probability to diagnose diseases and predict patient outcomes.
- Sports: Coaches and players use probability to make strategic decisions during games.
Final Thoughts
Q: What is the probability of randomly choosing a blue shirt from Sam's closet?
A: To find the probability of randomly choosing a blue shirt, we need to divide the number of blue shirts by the total number of shirts in the sample space. In this case, the number of blue shirts is 8, and the total number of shirts is 15. Therefore, the probability of randomly choosing a blue shirt is 8/15.
Q: What is the probability of randomly choosing a shirt with stripes from Sam's closet?
A: To find the probability of randomly choosing a shirt with stripes, we need to divide the number of shirts with stripes by the total number of shirts in the sample space. In this case, the number of shirts with stripes is 9 (5 blue shirts with stripes + 4 green shirts with stripes), and the total number of shirts is 15. Therefore, the probability of randomly choosing a shirt with stripes is 9/15.
Q: What is the probability of randomly choosing a blue shirt or a shirt with stripes from Sam's closet?
A: We already calculated this probability in the previous section. The probability of randomly choosing a blue shirt or a shirt with stripes is 3/5.
Q: How do we calculate the probability of an event occurring?
A: To calculate the probability of an event occurring, we need to follow these steps:
- Define the sample space, which is the set of all possible outcomes.
- Determine the number of favorable outcomes, which is the number of outcomes that meet the condition of the event.
- Divide the number of favorable outcomes by the total number of possible outcomes to find the probability.
Q: What is the difference between probability and chance?
A: Probability and chance are related but distinct concepts. Probability is a measure of the likelihood of an event occurring, while chance is the outcome of an event. For example, the probability of rolling a six on a fair six-sided die is 1/6, but the chance of rolling a six is the actual outcome of the roll.
Q: Can we use probability to predict the future?
A: While probability can help us understand the likelihood of events occurring, it is not a guarantee of the future. Probability is based on past data and trends, but it does not take into account unforeseen events or changes in circumstances.
Q: How do we use probability in real-life scenarios?
A: Probability is used in various real-life scenarios, such as:
- Insurance: Insurance companies use probability to determine the likelihood of an event occurring and set premiums accordingly.
- Finance: Financial institutions use probability to make investment decisions and manage risk.
- Medicine: Medical professionals use probability to diagnose diseases and predict patient outcomes.
- Sports: Coaches and players use probability to make strategic decisions during games.
Q: What are some common probability concepts?
A: Some common probability concepts include:
- Independent events: Events that do not affect each other's probability.
- Dependent events: Events that affect each other's probability.
- Mutually exclusive events: Events that cannot occur at the same time.
- Conditional probability: The probability of an event occurring given that another event has occurred.
Q: How do we calculate conditional probability?
A: To calculate conditional probability, we need to follow these steps:
- Define the sample space, which is the set of all possible outcomes.
- Determine the number of favorable outcomes, which is the number of outcomes that meet the condition of the event.
- Divide the number of favorable outcomes by the total number of possible outcomes to find the probability.
- Multiply the probability by the probability of the condition occurring to find the conditional probability.
Conclusion
In this article, we answered some frequently asked questions about probability, including how to calculate the probability of an event occurring, the difference between probability and chance, and how to use probability in real-life scenarios. We also discussed some common probability concepts, such as independent events, dependent events, mutually exclusive events, and conditional probability. By understanding these concepts, we can make informed decisions and predictions in various fields.