Probability: Events, Rules, Bayes’ Theorem, Conditional Probability

TAKEAWAYS:

• Probability measures how likely an event is to occur, ranging from 0 (impossible) to 1 (certain).
• The addition rule combines probabilities of two events, while the multiplication rule finds the probability of both events happening together.
• Conditional probability helps find the likelihood of an event given that another event has already occurred.
• Bayes’ theorem updates probabilities based on new information, widely used in medicine and machine learning.
• Probability is applied in real life, from predicting weather to calculating risks in games and finance.

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What is Probability?

Probability is a branch of mathematics that deals with the chance or likelihood of an event happening. It is measured on a scale from 0 to 1, where:

• 0 means the event cannot happen (impossible).
• 1 means the event will happen (certain).

For example, if you toss a fair coin, the probability of getting heads is 1/2 or 0.5, because there are two possible outcomes (heads or tails), and only one of them is heads.

Key Terms

1. Experiment: An action or process that leads to one or more outcomes (e.g., rolling a die, drawing a card).
2. Sample Space (S): The set of all possible outcomes of an experiment.
   • Example: When a die is rolled, the sample space is S = {1, 2, 3, 4, 5, 6}.
3. Event (E): A subset of the sample space. An event can be a single outcome or a group of outcomes.
   • Example: Getting an even number when rolling a die is the event E = {2, 4, 6}.

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Types of Events

Events can be classified into different types based on their relationship with the sample space and other events:

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Probability of an Event

The probability of an event E is given by: $$P(E)=\frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$$

Example 1: Probability of Rolling a Die

Problem: What is the probability of rolling an even number on a fair six-sided die?

Solution:

1. Sample Space (S): {1, 2, 3, 4, 5, 6}
2. Event (E = even numbers): {2, 4, 6}
3. Number of favorable outcomes: 3 (2, 4, 6)
4. Total number of possible outcomes: 6
5. Probability: $$P(E)=\frac{3}{6}=\frac{1}{2}=0.5$$

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Addition Rule of Probability

The addition rule helps find the probability of either of two events occurring. There are two cases:

1. Mutually Exclusive Events (A and B cannot happen together): $$P(A\text{ or }B)=P(A)+P(B)$$

2. Non-Mutually Exclusive Events (A and B can happen together): $$P(A\text{ or }B)=P(A)+P(B)-P(A\text{ and }B)$$

Example 2: Probability of Drawing a King or a Heart

Problem: A card is drawn from a standard deck of 52 cards. What is the probability that the card is a king or a heart?

Solution:

1. Total number of cards: 52
2. Number of kings (K): 4
3. Number of hearts (H): 13
4. Number of king of hearts (K ∩ H): 1 (since the king of hearts is counted in both K and H)
5. Probability: $$P(K\text{ or }H)=P(K)+P(H)-P(K\text{ and }H)=\frac{4}{52}+\frac{13}{52}-\frac{1}{52}=\frac{16}{52}=\frac{4}{13}$$

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Multiplication Rule of Probability

The multiplication rule finds the probability of both events occurring. There are two cases:

1. Independent Events (A and B are independent): $$P(A\text{ and }B)=P(A)\times P(B)$$

2. Dependent Events (A and B are dependent): $$P(A\text{ and }B)=P(A)\times P(B|A)$$ Here, $P(B|A)$ is the conditional probability of B given that A has already occurred.

Example 3: Probability of Two Coin Tosses

Problem: What is the probability of getting two heads in two successive tosses of a fair coin?

Solution:

1. Probability of first head (H₁): $\frac{1}{2}$
2. Probability of second head (H₂): $\frac{1}{2}$ (independent of the first toss)
3. Probability of both heads: $$P(H_1\text{ and }{H}_2)=P(H_1)\times P(H_2)=\frac{1}{2}\times \frac{1}{2}=\frac{1}{4}$$

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Conditional Probability

Conditional probability is the probability of an event B occurring given that another event A has already occurred. It is denoted as $P(B|A)$ and is calculated as: $$P(B|A)=\frac{P(A\text{ and }B)}{P(A)}$$

Example 4: Probability of Drawing Two Aces

Problem: Two cards are drawn without replacement from a standard deck of 52 cards. What is the probability that the second card is an ace, given that the first card was a king?

Solution:

1. Total cards initially: 52
2. First card drawn: King (not an ace), so remaining cards = 51
3. Number of aces remaining: 4 (since no ace was removed)
4. Probability of second card being an ace given first was a king: $$P(\text{Ace}|\text{King})=\frac{4}{51}$$

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Bayes’ Theorem

Bayes’ theorem relates the conditional and marginal probabilities of random events. It is used to update probabilities based on new information. The formula is: $$P(A|B)=\frac{P(B|A)\times P(A)}{P(B)}$$

Example 5: Medical Testing

Problem: A disease affects 1% of the population. A test for the disease is 95% accurate (i.e., it correctly identifies 95% of infected people and 95% of healthy people). If a randomly selected person tests positive, what is the probability that they actually have the disease?

Solution: Let:

• D = event of having the disease.
• ¬D = event of not having the disease.
• T⁺ = event of testing positive.

Given:

• $P(D)=0.01$
• $P(\neg D)=0.99$
• $P(T^{+}|D)=0.95$ (true positive rate)
• $P(T^{+}|\neg D)=0.05$ (false positive rate)

We need to find $P(D|T^{+})$.

Using Bayes’ theorem: $$P(D|T^{+})=\frac{P(T^{+}|D)\times P(D)}{P(T^{+})}$$ Where: $$P(T^{+})=P(T^{+}|D)\times P(D)+P(T^{+}|\neg D)\times P(\neg D)$$ $$P(T^{+})=(0.95\times 0.01)+(0.05\times 0.99)=0.0095+0.0495=0.059$$

Now, plug into Bayes’ theorem: $$P(D|T^{+})=\frac{0.95\times 0.01}{0.059}=\frac{0.0095}{0.059}\approx 0.161\text{ or }16.1\%$$

Interpretation: Even if a person tests positive, there is only a 16.1% chance they actually have the disease. This shows why false positives can be misleading!

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Applications of Probability

Probability is used in many real-life situations:

1. Games of Chance: Calculating odds in poker, roulette, or lotteries.
2. Medical Testing: Determining the accuracy of diagnostic tests (as in Example 5).
3. Weather Forecasting: Predicting the chance of rain.
4. Finance: Assessing risks in investments (e.g., stock market probabilities).
5. Quality Control: Manufacturing industries use probability to check defective products.
6. Sports: Calculating the probability of winning a match or tournament.

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Common Mistakes to Avoid

1. Assuming Independence: Always check if events are independent before multiplying probabilities.
2. Ignoring Sample Space: Forgetting to consider all possible outcomes can lead to incorrect probabilities.
3. Misapplying Bayes’ Theorem: Confusing $P(A|B)$ with $P(B|A)$.
4. Overlooking Conditional Probability: Not updating probabilities when events are dependent.

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Exam Tip

1. Understand Definitions: Know the difference between sample space, event, mutually exclusive, and independent events.
2. Practice Addition and Multiplication Rules: Always check if events are mutually exclusive or independent before applying the rules.
3. Draw Venn Diagrams: Visualizing events with Venn diagrams helps in solving complex probability problems.
4. Memorize Bayes’ Theorem: This is a high-weightage topic in exams. Practice problems involving medical testing or spam detection.
5. Show All Steps: In NEB exams, partial credit is given for correct steps, even if the final answer is wrong.
6. Real-World Problems: Many exam questions are based on real-life scenarios (e.g., cards, dice, medical tests). Relate the problem to what you know.

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NEB Board-Style Questions

Short Answer Questions

1. Define conditional probability with an example.
2. What is the difference between independent and dependent events?
3. State Bayes’ theorem and explain its importance.
4. If $P(A)=0.4$, $P(B)=0.3$, and $P(A\text{ and }B)=0.1$, find $P(A\text{ or }B)$.
5. Two dice are rolled. What is the probability that the sum is 7?

Long Answer Questions

6. A bag contains 5 red, 4 blue, and 3 green balls. Two balls are drawn without replacement. Find the probability that:

   • Both balls are red.
   • One ball is blue and the other is green.
   • Neither ball is green.

7. In a class of 40 students, 24 like math, 16 like science, and 8 like both. If a student is chosen at random, what is the probability that:

   • The student likes math or science?
   • The student likes neither math nor science?
   • The student likes math but not science?

8. A factory produces 10% defective items. If 5 items are inspected, what is the probability that:

   • Exactly 2 are defective?
   • At least 1 is defective?
   • None are defective?

9. A test for a disease has a 90% true positive rate and a 5% false positive rate. If 2% of the population has the disease, what is the probability that a person who tests positive actually has the disease? Use Bayes’ theorem.

10. Explain the addition rule and multiplication rule of probability with suitable examples. When would you use each rule?

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Note: Always practice problems from past NEB question papers to get familiar with the exam pattern. Probability questions often involve cards, dice, coins, or real-world scenarios, so relate the problem to these examples. Good luck!