Probability Formulas with Examples for Aptitude Problems

Probability Formulas with Examples for Aptitude Problems

Probability Formulas with Examples for Aptitude Problems

Are you ready to dive into the world of probability? Whether you’re preparing for an exam or simply looking to understand how probability works, this article will help you grasp the basic concepts, important formulas, and how to solve example problems effectively.

What is Probability?

Probability is the measure of the likelihood that a given event will occur. It is a mathematical concept that always ranges between 0 (impossible event) and 1 (certain event). In simple terms, the closer the probability is to 1, the more likely the event is to occur.

Important Terms in Probability

To fully understand probability, it’s essential to know these key terms:
  • Sample Space (S): The set of all possible outcomes of an experiment.
    • Example: In a coin toss, S = {H, T}. In a dice roll, S = {1, 2, 3, 4, 5, 6}.
  • Event: A subset of the sample space, consisting of one or more outcomes.
  • Mutually Exclusive Events: Two events are mutually exclusive if they cannot occur at the same time.
    • Example: Tossing a coin and getting both heads and tails is mutually exclusive.
  • Collectively Exhaustive Events: Events that together include all possible outcomes.
    • Example: When throwing a die, “getting an odd number” and “getting an even number” are collectively exhaustive events.
  • Independent Events: Two events are independent if the occurrence of one does not affect the probability of the other.
    • Example: Tossing a coin and rolling a die are independent events.

Key Probability Formulas

To solve probability problems quickly, it’s crucial to familiarize yourself with these fundamental formulas:
  • Probability of an Event (P(E)):P(E)=n(E)n(S)P(E) = \frac{n(E)}{n(S)}Where n(E) is the number of favorable outcomes and n(S) is the total number of outcomes in the sample space.
  • Probability of Sample Space (P(S)):P(S)=1P(S) = 1The probability of the sample space is always 1.
  • Probability of an Impossible Event (P(∅)):P(∅)=0P(∅) = 0The probability of an impossible event is 0.
  • Addition Rule: For two events A and B:P(A∪B)=P(A)+P(B)−P(A∩B)P(A \cup B) = P(A) + P(B) – P(A \cap B)If A and B are mutually exclusive:P(A∪B)=P(A)+P(B)P(A \cup B) = P(A) + P(B)
 

Theorems in Probability

1. Addition Theorem

The Addition Theorem is used to calculate the probability of either event A or event B happening. If A and B are mutually exclusive, the probability of A or B is simply the sum of their probabilities.
  • Formula: P(A∪B)=P(A)+P(B)−P(A∩B)P(A \cup B) = P(A) + P(B) – P(A \cap B)
Example 1: A stationery box contains 3 red paper clips, 4 green paper clips, and 5 blue paper clips. If one paper clip is taken and then replaced, what’s the probability that the first paper clip is red and the second one is blue?Solution:P(red and blue)=P(red)×P(blue)=312×512=548P(\text{red and blue}) = P(\text{red}) \times P(\text{blue}) = \frac{3}{12} \times \frac{5}{12} = \frac{5}{48}

2. Multiplication Theorem

For independent events, the probability of both events happening is the product of their individual probabilities.
  • Formula: P(A∩B)=P(A)×P(B)P(A \cap B) = P(A) \times P(B)
Example 2: What is the probability of selecting either a queen or a heart from a deck of 52 cards?Solution:P(queen or heart)=452+1352−152=1652=413P(\text{queen or heart}) = \frac{4}{52} + \frac{13}{52} – \frac{1}{52} = \frac{16}{52} = \frac{4}{13} 

Conditional Probability

Conditional probability refers to the probability of an event occurring given that another event has already occurred.
  • Formula: P(A∣B)=P(A∩B)P(B)P(A \mid B) = \frac{P(A \cap B)}{P(B)}
Example 3: If the probability of it being Tuesday and Raman being absent is 0.03, and the probability of it being Tuesday is 0.2, what’s the probability that Raman is absent given it’s Tuesday?Solution:P(Absent∣Tuesday)=0.030.2=0.15P(\text{Absent} \mid \text{Tuesday}) = \frac{0.03}{0.2} = 0.15 

Expected Value

Expected value helps you determine the average outcome when an experiment is repeated many times. It’s calculated by multiplying each outcome by its probability and then summing up the results.
  • Formula: E(X)=∑(P(x)×x)E(X) = \sum (P(x) \times x)
Example 4: In a gambling game, you bet on the roll of a die. The casino pays Rs. 120 for a 6 and charges Rs. 30 for any other number. The expected value is calculated as follows:E(G)=16×120+56×(−30)=−5E(G) = \frac{1}{6} \times 120 + \frac{5}{6} \times (-30) = -5Thus, over time, the casino will benefit by Rs. 5 per game. 

Conclusion: How to Master Probability

Now that you’ve learned the core concepts and formulas in probability, it’s time to put them into practice. Here’s how you can prepare:
  1. Practice Problem-Solving: The more problems you solve, the better your understanding will be.
  2. Use Probability in Real-Life Scenarios: Apply these concepts to games, weather forecasting, or financial planning to see how probability works in action.
  3. Review Key Theorems: Master the addition and multiplication theorems for solving complex problems quickly.
By applying these concepts and formulas, you can confidently solve probability problems and even ace related exams
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