Conditional Probability and Independence in Discrete Mathematics

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Conditional Probability and Independence are the concepts in Probability Theory that help us to find out the likelihood of an event occurring, given that something else has already occurred. In this chapter, we will explain what Conditional Probability and Independence means in Probability.

What is Conditional Probability?

In simple terms, Conditional Probability is the probability of an event that occurs given that another event has already occurred.

Instead of looking at the whole sample space (all the possible outcomes), we are focusing on a specific part of it. The probability of one event happening changes because we know something else has already occurred.

We write it as , which is read as "the probability of event E given that event F has occurred." The formula for conditional probability looks like this −

$$\mathrm{P(E \:\mid\: F) \:=\: \frac{P(E \:\cap\: F)}{P(F)}}$$

Let's understand this formula −

• P(E ∩ F) − It is the probability that both events E and F Like finding the overlap between two circles in a Venn diagram.
• P(F) − This is just the probability of event F And It is important that P(F) ≠ 0, because we cannot divide by zero.

So, conditional probability is basically the ratio of the chance that both E and F happen to the chance that F occurs alone.

Example of Conditional Probability: College Graduates and Salaries

Imagine in ABC Company. About 25% of the employees earn more than Rs.50,000 a year. Also, 20% of the employees are college graduates, and 15% of the employees are college graduates earning more than Rs.50,000 a year.

Now, consider we want to find the probability that a randomly selected employee earns more than Rs.50,000. It is given that they are a college graduate.

In other words, we are trying to figure out P(H C), where:

• H is the event that the employee earns more than Rs.50,000.
• C is the event that the employee is a college graduate.

From the problem, we know the following −

• P(H) = 0.25 (25% earn more than Rs.50,000).
• P(C) = 0.20 (20% are college graduates).
• P(H ∩ C) = 0.15 (15% are college graduates earning more than Rs.50,000).

Using the conditional probability formula −

$$\mathrm{P(H \:\mid\: C) \:=\: \frac{P(H \:\cap\: C)}{P(C)} \:=\: \frac{0.15}{0.20} \:=\: 0.75}$$

So, the probability that an employee earns more than Rs.50,000. This given that they are a college graduate, is 75%. So, if someone is a college graduate, there is a high chance that they earn more than Rs.50,000.

Conditional Probability with Equally Likely Outcomes

Conditional probability gets a bit simpler when all the outcomes of an experiment are equally likely. In this case, instead of solving with probabilities directly, we can just count the number of outcomes.

The formula changes slightly to −

$$\mathrm{P(E \:\mid\: F) \:=\: \frac{\text{Number of outcomes in } E \:\cap\: F}{\text{Number of outcomes in } F}}$$

Example: Picking Balls from an Urn

Consider we have an urn with 8 white balls and 2 green balls. If we randomly pick two balls, what is the probability that the second ball is white? If it is given that the first ball was white?

We start by looking at how many outcomes match our condition. If the first ball is white, there are 8 possibilities for the first ball. After removing that white ball, there are 7 white balls left and 9 balls in total. So, there are 56 ways for both the first and second balls to be white.

Since there are 72 possible ways to pick the second ball after a white ball was picked first, the conditional probability is −

$$\mathrm{P(\text{second ball is white} \:\mid\: \text{first ball is white}) \:=\: \frac{56}{72} \:=\: \frac{7}{9}}$$

So, there are 7/9 chance that the second ball is white, when first one was white.

The Product Rule

The product rule are also used in conditional probability. It helps us to calculate the probability of both events E and F happening together. The formula looks like this −

$$\mathrm{P(E \:\cap\: F) \:=\: P(F) \:\times\: P(E \:\mid\: F)}$$

This rule is just a rearrangement of the conditional probability. But it is useful in situations where we want to find the chance that two things happen together.

Example: Drawing Cards from a Deck

Let us say we are drawing two cards from a standard deck of 52 cards, without replacement. What will be the probability that the first card is red and the second card is black? Let,

• F be the event that the first card is red.
• E be the event that the second card is black.

We know that the probability of drawing a red card first is 26/52 = 0.5, since half of the deck is red cards.

After drawing a red card, there are 26 black cards and 51 cards left in total. So the conditional probability that the second card is black, given that the first card was red, is 26/51.

Using the product rule:

$$\mathrm{P(E \:\cap\: F) \:=\: P(F) \:\times\: P(E \:\mid\: F)\: =\: \frac{1}{2} \:\times\: \frac{26}{51} \:=\: \frac{13}{51}}$$

So, the probability of drawing a red card first and a black card second is about 25%.

Independence in Probability

Independence means something like "unrelated" or "separate.", but in probability, it's a bit more specific. Two events E and F are independent if knowing that one event occurred does not change the probability of the other event.

In other words, E and F are independent if −

$$\mathrm{P(E \:\cap\: F) \:=\: P(E) \:\times\: P(F)}$$

Or equivalently, if:

• P(E | F) = P(E)
• And, P(F | E) = P(F)

Example: Brown Eyes and Majors

Let us consider we randomly pick a student, and we are interested in two events:

• E − The student is a psychology major.
• F − The student has brown eyes.

These two events are independent. Whether someone has brown eyes does not affect whether they are a psychology major or not. So, knowing someone's eye color does not change the probability that they are a psychology major.

In this case, P(E | F) = P(E), (they are independent). Independence in probability is about whether events influence each other, not whether they are physically separate or unrelated in a real-world sense.

Independence of Multiple Events

When we have more than two events, like independence of a set of events. For example, if we have three events E₁, E₂, and E₃, we say they are independent if the occurrence of one event does not affect the others.

$$\mathrm{P(E_1 \:\cap\: E_2 \:\cap\: E_3) \:=\: P(E_1) \:\times\: P(E_2) \:\times\: P(E_3)}$$

Example: Stereo System Defects

Let us consider a company manufactures stereo systems. Quality control shows that 2% of CD players, 3% of amplifiers, and 7% of speakers are defective. What are the probability that a stereo system (which includes one CD player, one amplifier, and two speakers) is not defective?

The probability that each component is not defective is −

• CD player: 0.98
• Amplifier: 0.97
• Speaker: 0.93 (and we have two speakers, so we square this)

Since defects are independent, we multiply these probabilities together:

$$\mathrm{P(\text{non defective}) \:=\: 0.98 \:\times\: 0.97 \:\times\: 0.93^2 \:\approx\: 0.82}$$

So, the probability that the stereo system is not defective is about 82%.

Conclusion

In this chapter, we explained the concepts of conditional probability and independence in probability theory. We also explored the Product Rule and learned how to calculate the likelihood of multiple events happening together. Finally, we covered what it means for events to be independent and how they are useful in real-world situations.