The application of probability can be seen in quantitative aptitude as well as in daily life. It is essential to understand the basic concepts of probability to solve the problems efficiently.
Probability determines how likely an event is to occur. It is represented as a numeric value ranging from 0 to 1. This article will cover the foundational concepts of probability and provide techniques to solve problems of varying difficulty levels, helping students prepare for competitive exams and placements.
Probability Formula
Probability = \dfrac{Favorable \ Outcome}{Total \ Number \ of \ Outcomes}
Tips and Tricks for Probability
1) When solving probability problems, if it's easier to calculate the probability of an event not occurring, subtract that probability from 1 to find the likelihood of the event happening. For example, P(E’) = 1 – P(E).
2) If the term "or" appears in a question, use addition (+) when applying the Fundamental Principle of Counting.
3) If the term "and" appears in a question, use multiplication (×) when applying the Fundamental Principle of Counting.
4) Number of Outcomes for Rolling Dice = 6n, where n is the number of dice rolled at once:
- For 1 die, the outcomes are 61 = 6.
- For 2 dice, the outcomes are 62 = 36.
- For 3 dice, the outcomes are 63 = 216.
5) Number of Outcomes for Tossing Coins = 2n, where n is the number of coins tossed at once:
- For 1 coin, the outcomes are 21 = 2.
- For 2 coins, the outcomes are 22 = 4.
- For 3 coins, the outcomes are 23 = 8.
6) Number of Outcomes for Drawing a Card = 52n, where n is the number of sets or decks of cards used:
- For 1 deck of cards, the outcomes are 521 = 52.
- For 2 decks of cards, the outcomes are 522 = 2,704.
- For 3 decks of cards, the outcomes are 523 = 140,608.
7) The sum of the probability of an event and its complement is 1, i.e., P(A) + P(A′) = 1.
8) The probability of an impossible event or an event not happening is always 0, i.e., P(∅) = 0.
9) The probability of a sure event is always 1, i.e., P(A) = 1.
10) The probability of any event lies between 0 and 1, i.e., 0 ≤ P(A) ≤ 1.
11) The formula for the union of two events A and B: P(A∪B) = P(A) + P(B) − P(A∩B).
12) For mutually exclusive events A and B: P(A∪B) = P(A) + P(B).
13) Additional Formulas:
- Conditional Probability: P(A∣B) =P(A∩B)/ P(B) , where P(B) ≠ 0
- Probability of Intersection of Two Events: P(A∩B) = P(A) × P(B∣A).
14) Use Venn Diagrams: For problems involving intersections or unions of events (like "either A or B happens"), a Venn diagram can help visualize the problem and ensure that you're not double-counting or overlooking possibilities.
Using the above Venn diagram, let the universal set be U = {1,2,3,4,5,6,8}, A = {1,2,3,4,5,6}, and B = {2,4,6,8}.
From the diagram we can clearly see:
- A − B = {1,3,5} (elements only in A)
- A ∩ B = {2,4,6} (elements common to both A and B)
- B − A = {8} (elements only in B)
If we want A ∪ B (either A or B happens), we take all elements in both circles: A ∪ B = {1,2,3,4,5,6,8}.
15) When a coin is tossed n times or n coins are tossed once, the probability of each specific outcome is 1/2n.
16) When a die is rolled n times or n dice are rolled simultaneously, the probability of any specific outcome is 1/6n.
17) If n cards are drawn from a deck without replacement, the number of possible outcomes is determined by combinations. The probability of each specific event is given by:
- P (E) = 1/C(52, n)
18) If n cards are drawn one after the other with replacement, each draw is independent and the number of possible outcomes for each draw remains 52. Therefore, the probability of each simple event is given by:
- P(E) = 1/52n.
Note: A probability of 1 means an event will definitely occur, while 0 means it will definitely not occur.'
Probability - Questions and Answers
Question 1 : Three unbiased coins are tossed. What is the probability that atmost one head occurs ?
Solution :
S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}
Favorable outcomes = {HTT, THT, TTH, TTT}
Total number of outcomes = 8
Number of favorable outcomes = 4
Required probability = 4 / 8 = 0.50
Question 2 : Find the probability of getting a red card when a card is drawn from a well shuffled pack of cards.
Solution :
Total number of outcomes = 52
Number of favorable outcomes = Number of red cards = 26
=> Required probability = 26 / 52 = 0.50
Question 3 : A bag contains 6 white and 4 black balls. Two balls are drawn at random from the bag. Find the probability that both the balls are of the same color.
Solution :
Outcome will be favorable if the two balls drawn are of the same color.
=> Number of favorable outcomes =
6C2 + 4C2 = 21
Total number of outcomes = 10C2 = 45
Therefore, required probability = 21 / 45 = 7 / 15
Question 4 : An unbiased die is tossed. Find the probability of getting an even number.
Solution :
S = {1, 2, 3, 4, 5, 6}
Favorable outcomes = {2, 4, 6}
Required probability = 3 / 6 = 0.50
Question 5 : From a bag containing red and blue balls, 10 each, 2 balls are drawn at random. Find the probability that one of them is red and the other is blue.
Solution :
Total number of outcomes = 20C2 = 190
Number of favorable outcomes = 10C1 x 10C1 = 100
Therefore, required probability = 100 / 190 = 10 / 19