Probability

Definition

Conditional probability is the probability of an event occurring, given that another event has already happened. It measures how new information updates the likelihood of an outcome.

Explanation

When we have "given" information, the total number of possible outcomes (the sample space) usually shrinks. We are no longer looking at the whole world of possibilities, but only the part where the condition is true.

• Notation: $P(A|B)$ is read as "the probability of A given B."
• Formula: $P(A|B) = \frac{P(A \cap B)}{P(B)}$

Examples

• Drawing Cards: What is the probability of drawing a King, given that you know the card drawn is a Face card? (The sample space shrinks from 52 cards to just the 12 face cards).
• Exam Results: The probability that a student passes the math exam, given that they attended all the revision classes.

Real-Life Use

Medical Diagnosis: Doctors use conditional probability to interpret test results. A doctor calculates the probability of a disease given a positive test result, factoring in the disease's prevalence in the general population.

Meaning

The Multiplication Theorem helps find the probability of two events happening in sequence. Independent events are those where the first outcome has zero impact on the second.

Key Points

• Independent Events: $P(A \cap B) = P(A) \times P(B)$.
• Dependent Events: $P(A \cap B) = P(A) \times P(B|A)$.

Examples

• Dice Rolls: Rolling a 6 and then rolling another 6. These are independent. Probability = $(1/6) \times (1/6) = 1/36$.
• Picking Marbles (Without Replacement): If you pick a red marble and keep it, the next pick is dependent because the total count of marbles has changed.

Real-Life Use

Quality Control: If a machine has a 1% error rate, engineers multiply the error rates to find the probability of two consecutive items being faulty ($0.01 \times 0.01$).

Definition

Total Probability is the sum of probabilities of an event happening through all possible different paths. Bayes' Theorem allows us to "reverse" the probability—finding the probability of a cause given an observed effect.

Formulas

• Total Probability: $P(A) = \sum P(A|B_i) \cdot P(B_i)$
• Bayes' Theorem: $P(B|A) = \frac{P(A|B) \cdot P(B)}{P(A)}$

Examples

• Email Spam Filters: A filter calculates the probability an email is "Spam" given that it contains specific words like "Free" or "Winner."
• Factory Machines: If a defective part is found, Bayes' Theorem identifies which of the three machines (A, B, or C) most likely caused the defect.

Real-Life Use

Spam Detection: This is the backbone of Gmail/Outlook. It calculates the likelihood of a message being spam based on the "evidence" of the words it contains.

Meaning

A Random Variable is a numerical description of an outcome. A Probability Distribution is a list or formula showing all possible values and their respective likelihoods.

Types

• Discrete Random Variable: Values you can count (e.g., number of children in a family).
• Continuous Random Variable: Values you measure (e.g., the exact height of a person or time spent waiting).

Examples

• Discrete: Number of siblings (0, 1, 2...).
• Continuous: The time you wait for a bus (e.g., 5.22 minutes).

Real-Life Use

Insurance: Companies treat "number of accidents per year" as a random variable to calculate premiums.

Definition

A Bernoulli Trial is a single experiment with only two outcomes: Success or Failure. The Binomial Distribution describes the outcome of repeating these trials $n$ times.

Formula

$$P(X = r) = \binom{n}{r} \cdot p^r \cdot q^{n-r}$$

Where:

• $n$ = total trials
• $r$ = number of successes
• $p$ = probability of success
• $q$ = probability of failure ($1-p$)

Examples

• Coin Toss: Flipping a coin 10 times and finding the chance of exactly 7 heads.
• Product Testing: Checking 50 lightbulbs to see how many are defective.

Real-Life Use

Clinical Drug Trials: Calculating the probability that out of 100 patients, a specific number will recover based on the drug's known success rate.

Meaning

The Mean (Expected Value) is the average outcome you'd expect over many trials. Variance measures how much the outcomes spread out from that average.

Formulas

• Mean (Expected Value) $E(X)$: $\sum [x \cdot P(x)]$
• Variance $Var(X)$: $E(X^2) - [E(X)]^2$

Examples

• Lottery: The expected value of a \$2 ticket is usually less than \$2, explaining why the "house" wins.
• Stock Market: A stock with high variance is considered "risky" due to price volatility.

Problem 1: Bayes' Theorem

A factory has two machines, A and B. Machine A produces 60% of the items and Machine B produces 40%. 2% of items from A are defective, and 5% of items from B are defective. If an item is chosen at random and found to be defective, what is the probability it came from Machine B?

1. Identify Probabilities:
   P(A) = 0.60, P(B) = 0.40
   P(D|A) = 0.02, P(D|B) = 0.05

2. Total Probability of Defective P(D):
   P(D) = [P(A) * P(D|A)] + [P(B) * P(D|B)]
   P(D) = (0.60 * 0.02) + (0.40 * 0.05) = 0.032

3. Apply Bayes' Theorem for P(B|D):
   P(B|D) = [P(B) * P(D|B)] / P(D)
   P(B|D) = 0.020 / 0.032 = 0.625
   
Answer: 62.5%
        

Problem 2: Binomial Distribution

A marksman has a probability of 0.8 of hitting a target. If he fires 5 shots, what is the probability he hits exactly 3 targets?

1. Variables: n=5, r=3, p=0.8, q=0.2
2. Formula: P(X=3) = C(5,3) * (0.8)^3 * (0.2)^2
3. Calculation:
   C(5,3) = 10
   (0.8)^3 = 0.512
   (0.2)^2 = 0.04
4. Result: 10 * 0.512 * 0.04 = 0.2048

Answer: 20.48%
        

1. If $P(A) = 0.3$ and $P(B) = 0.4$, and A and B are independent, what is $P(A \cap B)$?
   Answer: c) 0.12
2. The probability of an event that is certain to happen is:
   Answer: c) 1
3. Which theorem is used to find $P(\text{Cause}|\text{Effect})$?
   Answer: b) Bayes' Theorem
4. A Bernoulli trial has how many outcomes?
   Answer: b) 2
5. If two events are mutually exclusive, $P(A \cap B)$ is:
   Answer: c) 0

1. Q: Define Conditional Probability and provide the formula.
   Ans: It is the probability of an event occurring given another has occurred. Formula: $P(A|B) = P(A \cap B) / P(B)$.
2. Q: Differentiate between Independent and Mutually Exclusive events.
   Ans: Independent events do not affect each other's occurrence ($P(A|B)=P(A)$). Mutually exclusive events cannot happen at the same time ($P(A \cap B)=0$).
3. Q: Can a probability be negative? Why?
   Ans: No, probability is defined on the interval $[0, 1]$. A negative value has no physical meaning in measuring likelihood.

• B-I-N-S (Binomial): Both outcomes, Independent, Number of trials fixed, Same probability.
• Bayes: "Bayes flips the condition."
• Mean: The center. Variance: The spread.