An experiment has 10 equally likely outcomes. Let A and B be non-empty events of the experiment. If

Q: Wait, why do we reject the negative root here?

In this problem, we are dealing with the number of outcomes in an event, which is a count. A count cannot be negative. Therefore, we only consider positive integer values for $x$ and $y$.

Q: Why can't we just use energy conservation?

This is a probability problem, not a mechanics problem! Energy conservation applies to physical systems. Here, we are working with the logic of sets and independent events. The 'energy' here is the consistency of the probability axioms.

Q: Is it possible for $x$ to be something other than a multiple of 5?

Great question! Look at the equation $y = \frac{2x}{5}$. For $y$ to be an integer, $2x$ must be divisible by 5. Since 2 and 5 are coprime, $x$ must be a multiple of 5. There is no other way to satisfy the integer constraint.

Question

An experiment has 10 equally likely outcomes. Let $A$ and $B$ be non-empty events of the experiment. If $A$ consists of 4 outcomes, the number of outcomes that $B$ must have so that $A$ and $B$ are independent, is

Options:

(A) 2, 4 or 8

(B) 3, 6 or 9

(C) 4 or 8

(D) 5 or 10

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing before a vast, empty canvas representing our sample space, a set of ten equally likely outcomes. This is our universe, $S$, where $n(S) = 10$.

Within this universe, we have an event $A$ that claims four of these outcomes for itself. We know $n(A) = 4$, which gives us a probability:

$$P(A) = \frac{4}{10} = \frac{2}{5}$$

Now, we introduce a mysterious event $B$. We do not know its size, so let us call it $x$, meaning $n(B) = x$. We also have an intersection, the shared territory between $A$ and $B$, which we will call $y$, so $n(A \cap B) = y$.

### The Heart of Independence

What does it truly mean for two events to be independent? It means that the occurrence of one provides absolutely no information about the occurrence of the other.

Mathematically, this is expressed as the elegant condition:

$$P(A \cap B) = P(A) \cdot P(B)$$

This is the heartbeat of our problem. Let us translate this into the language of our counts. We know $P(A \cap B) = \frac{y}{10}$, $P(A) = \frac{4}{10}$, and $P(B) = \frac{x}{10}$.

Substituting these into our independence condition, we get:

$$\frac{y}{10} = \frac{4}{10} \cdot \frac{x}{10}$$

### The Algebraic Journey

Now, let us simplify this. Multiplying both sides by 10, we find:

$$y = \frac{4x}{10} = \frac{2x}{5}$$

This is a beautiful, simple relationship. However, we must respect the constraints of our reality. Since $y$ is the number of outcomes in the intersection of two sets, it must be an integer.

For $y = \frac{2x}{5}$ to be an integer, $2x$ must be divisible by 5. Since 2 and 5 share no common factors, $x$ itself must be a multiple of 5.

### The Final Revelation

We know that $B$ is a non-empty event, so $x > 0$. We also know that $B$ is a subset of our sample space of ten, so $x \le 10$.

The multiples of 5 in the range $0 < x \le 10$ are exactly 5 and 10. Let us test them:

If $x = 5$, then $y = \frac{2(5)}{5} = 2$, which is a valid integer.

If $x = 10$, then $y = \frac{2(10)}{5} = 4$, which is also a valid integer.

Thus, the possible values for the number of outcomes in $B$ are **5 and 10**. We have navigated the constraints, respected the definitions, and arrived at the truth.

Animated Solution for Mathematics - Probability: An experiment has 10 equally likely outcomes. Let $A$ and $B$ be non-empty events of the experiment. If $A$ consists of 4 outcomes, the number of outcomes that $B$ must have so that $A$ and $B$ are independent, is

Select Answer:

Visualized Solution

Defining the Sample Space

Event $A$ and its Probability

Defining Event $B$ and Intersection

The Condition for Independence

Setting up the Equation

Simplifying the Relation

Applying Integer Constraints

Bounding the Values of $x$

Testing the First Multiple

Testing the Second Multiple

The Final Answer

The Sigma Insight: Addition and Multiplication Theorems

Analyzing the Setup

The Heart of Independence

The Algebraic Journey

The Final Revelation

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