Five Balls Numbered From 1 To 5 Are Placed Into A Bag. Some Are Grey And Some Are White. The Balls Numbered 2, 4, And 5 Are Grey, While The Balls Numbered 1 And 3 Are White. A Ball Is Selected At Random.Let $X$ Be The Event That The Selected

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Introduction

Probability is a fundamental concept in mathematics that deals with the study of chance events. It is a measure of the likelihood of an event occurring, and it is often expressed as a number between 0 and 1. In this article, we will explore the concept of probability through a simple example involving the selection of colored balls from a bag.

The Problem

Five balls numbered from 1 to 5 are placed into a bag. Some are grey and some are white. The balls numbered 2, 4, and 5 are grey, while the balls numbered 1 and 3 are white. A ball is selected at random. Let $X$ be the event that the selected ball is white.

Defining the Sample Space

The sample space is the set of all possible outcomes of an experiment. In this case, the sample space consists of the five balls numbered from 1 to 5. We can represent the sample space as:

S={1,2,3,4,5}S = \{1, 2, 3, 4, 5\}

Defining the Event

The event $X$ is defined as the selection of a white ball. We can represent the event $X$ as:

X={1,3}X = \{1, 3\}

Calculating the Probability

The probability of an event is calculated as the number of favorable outcomes divided by the total number of possible outcomes. In this case, the number of favorable outcomes is 2 (the selection of a white ball), and the total number of possible outcomes is 5 (the selection of any ball). Therefore, the probability of the event $X$ is:

P(X)=25P(X) = \frac{2}{5}

Interpretation of the Probability

The probability of the event $X$ is 2/5, which means that the likelihood of selecting a white ball is 2 out of 5, or 40%. This is a measure of the chance of the event occurring, and it can be used to make informed decisions in a variety of situations.

Calculating the Probability of the Complement

The complement of an event is the set of all outcomes that are not in the event. In this case, the complement of the event $X$ is the selection of a grey ball. We can represent the complement of the event $X$ as:

Xc={2,4,5}X^c = \{2, 4, 5\}

The probability of the complement of the event $X$ is:

P(Xc)=35P(X^c) = \frac{3}{5}

Understanding Conditional Probability

Conditional probability is a measure of the likelihood of an event occurring given that another event has occurred. In this case, we can calculate the conditional probability of the event $X$ given that the ball selected is numbered 1 or 3.

Let $Y$ be the event that the ball selected is numbered 1 or 3. We can represent the event $Y$ as:

Y={1,3}Y = \{1, 3\}

The conditional probability of the event $X$ given that the event $Y$ has occurred is:

P(X∣Y)=P(X∩Y)P(Y)P(X|Y) = \frac{P(X \cap Y)}{P(Y)}

Since the event $X$ and the event $Y$ are the same, we have:

P(X∣Y)=P(X)P(Y)P(X|Y) = \frac{P(X)}{P(Y)}

P(X∣Y)=2525P(X|Y) = \frac{\frac{2}{5}}{\frac{2}{5}}

P(X∣Y)=1P(X|Y) = 1

This means that if the ball selected is numbered 1 or 3, the probability of the event $X$ is 1, or 100%.

Conclusion

Introduction

In our previous article, we explored the concept of probability through a simple example involving the selection of colored balls from a bag. We defined the sample space, the event, and the probability of the event, and we calculated the probability of the complement of the event. We also discussed the concept of conditional probability and how it can be used to make informed decisions in a variety of situations.

Q&A

Q: What is the probability of selecting a white ball from the bag? A: The probability of selecting a white ball from the bag is 2/5, or 40%.

Q: What is the probability of selecting a grey ball from the bag? A: The probability of selecting a grey ball from the bag is 3/5, or 60%.

Q: What is the complement of the event X? A: The complement of the event X is the selection of a grey ball. We can represent the complement of the event X as:

Xc={2,4,5}X^c = \{2, 4, 5\}

Q: What is the probability of the complement of the event X? A: The probability of the complement of the event X is:

P(Xc)=35P(X^c) = \frac{3}{5}

Q: What is conditional probability? A: Conditional probability is a measure of the likelihood of an event occurring given that another event has occurred. In this case, we can calculate the conditional probability of the event X given that the ball selected is numbered 1 or 3.

Q: How do you calculate conditional probability? A: To calculate conditional probability, we use the formula:

P(X∣Y)=P(X∩Y)P(Y)P(X|Y) = \frac{P(X \cap Y)}{P(Y)}

Q: What is the conditional probability of the event X given that the ball selected is numbered 1 or 3? A: Since the event X and the event Y are the same, we have:

P(X∣Y)=P(X)P(Y)P(X|Y) = \frac{P(X)}{P(Y)}

P(X∣Y)=2525P(X|Y) = \frac{\frac{2}{5}}{\frac{2}{5}}

P(X∣Y)=1P(X|Y) = 1

This means that if the ball selected is numbered 1 or 3, the probability of the event X is 1, or 100%.

Q: What is the importance of understanding probability? A: Understanding probability is important in a variety of situations, including:

  • Making informed decisions in business and finance
  • Analyzing data and making predictions
  • Understanding the likelihood of events occurring
  • Making decisions in situations where there is uncertainty

Conclusion

In this article, we have answered some common questions about probability and how it is used in a variety of situations. We have discussed the concept of conditional probability and how it can be used to make informed decisions in situations where there is uncertainty. We hope that this article has been helpful in understanding the concept of probability and how it is used in real-world situations.