Question

A box contain 4 red and 3 black ball. One ball is taken away from the box. After that two balls are drawn at random and both found red, what is the probability that the first ball taken always was also red? (a) \((2 / 5)\) (b) \((4 / 7)\) (c) \((24 / 105)\) (d) None

Short answer

The probability that the first ball taken away always was also red given that the two balls drawn afterward were both red is \(\frac{2}{5}\), or option (a).

Step-by-step solution

Define the events

Let A be the event that the first ball taken away is red, and let B be the event that the two balls drawn afterward are both red.

Find the probability of A

There are a total of 7 balls, 4 red and 3 black. Therefore, the probability of the first ball (A) being red is given by: \(P(A) = \frac{4}{7}\)

Find the probability of B given A

If the event A occurs, we are left with 3 red and 3 black balls. The probability of drawing two red balls (B) given that the first ball taken away is red (A) can be calculated by: \( P(B | A) = \frac{3}{6} \times \frac{2}{5} = \frac{1}{5} \)

Use Bayes' theorem to find the conditional probability

We can now use Bayes' theorem to find the conditional probability of A given B. Bayes' theorem states that: \(P(A | B) = \frac{P(B | A) \times P(A)}{P(B)}\) However, we need to compute P(B) first. This is the probability of drawing two red balls without considering which ball was drawn first, and can be calculated as: \(P(B) = \frac{C(4,2)}{C(7,2)} = \frac{6}{21} = \frac{2}{7}\) Now we can use Bayes' theorem to find P(A | B): \(P(A | B) = \frac{P(B | A) \times P(A)}{P(B)} = \frac{\frac{1}{5} \times \frac{4}{7}}{\frac{2}{7}} = \frac{4}{10} = \frac{2}{5}\)

State the answer

So, the probability that the first ball taken away always was also red (A) given that the two balls drawn afterward were both red (B) is \(\frac{2}{5}\), or option (a).

Key concepts

Bayes' Theorem

Bayes' Theorem is a fundamental concept in probability theory that relates the likelihood of an event based on prior knowledge of conditions that might relate to the event. It allows us to update our beliefs about the probability of an event A, given the occurrence of another related event B.

Mathematically, Bayes' theorem is expressed as:\[\begin{equation}P(A | B) = \frac{P(B | A) \times P(A)}{P(B)}\end{equation}\]where:

• P(A | B) - the probability of event A given that B has occurred.
• P(B | A) - the probability of event B given that A has occurred.
• P(A) - the probability of event A occurring on its own.
• P(B) - the probability of event B occurring on its own.

In the context of our example involving the red and black balls, we used Bayes' theorem to deduce the likelihood that the first ball removed was red, given that the subsequent two drawn balls were red. While the step-by-step solution provided an answer, it can be beneficial to visualize the theorem through diagrams or probability trees to aid understanding.

Probability Theory

Probability theory is the branch of mathematics that deals with the likelihood of events occurring. It is used to predict outcomes in various fields, such as finance, science, and engineering. Understanding probability can help us make informed decisions in the face of uncertainty.

There are several key concepts within probability theory that are exemplified in our exercise:

• Sample Space - This is the set of all possible outcomes. In the case of the balls, it includes all the possible combinations of balls that could be drawn.
• Events - These are outcomes or combinations of outcomes from the sample space. For instance, drawing two red balls is an event.
• Conditional Probability - This is the probability of an event occurring given that another event has already occurred. The exercise we solved relied heavily on this concept.

By understanding these components and their relations, students can better grasp the nature of probability and improve their problem-solving skills. It's essential for students to practice with different types of probability problems to master these concepts.

Moreover, explaining the concepts with real-world scenarios can anchor the abstract mathematic theory to tangible examples, making it easier for students to understand and remember.

Combinatorics

Combinatorics is the field of mathematics focused on counting, both in a concrete sense - as in how many ways can a certain outcome occur - and in an abstract sense pertaining to certain properties of sets and finite structures. This field provides the mathematical underpinning for calculating probabilities when events have multiple outcomes.

Application in Probability

Combinatorics comes into play in probability when determining how many ways specific events can occur. The combination, which is one of the most essential concepts of combinatorics, was used in our exercise to find the number of ways to choose two red balls from a set of four.For instance, the combination formula is given by:\[\begin{equation}C(n, k) = \frac{n!}{k!(n-k)!}\end{equation}\]where:

• C(n, k) - the number of ways to choose k items from a set of n distinct items without regard to the order of selection.
• n! - the factorial of n, which is the product of all positive integers less than or equal to n.

Understanding these basic principles of combinatorics enables students to tackle a broad spectrum of problems within probability theory and is essential for deeper study in many areas of mathematics and science. Engaging in exercises that involve calculating permutations and combinations can provide practical experience with combinatorics principles.