Probability

Probability questions involving complementary and compound events are common in SSC CGL. Mastering these concepts with speed is key.

Core Formula

1. Complementary Event: The probability that an event E does not occur is P(E') = 1 - P(E). This is extremely useful for "at least one" type problems.

2. Compound Events (Addition Rule - 'OR'):

   • General: For any two events A and B, the probability that A or B occurs is P(A U B) = P(A) + P(B) - P(A ∩ B). Here, P(A ∩ B) is the probability that both A and B occur.
   • For Mutually Exclusive Events: If A and B cannot happen at the same time (e.g., drawing a King and an Ace in a single draw), then P(A ∩ B) = 0. So, P(A U B) = P(A) + P(B).

3. Compound Events (Multiplication Rule - 'AND'):

   • General (Dependent Events): The probability that A and B occur is P(A ∩ B) = P(A) * P(B|A), where P(B|A) is the conditional probability of B occurring given that A has already occurred.
   • For Independent Events: If the occurrence of A does not affect the occurrence of B (e.g., tossing two coins), then P(B|A) = P(B). So, P(A ∩ B) = P(A) * P(B).

Worked Example 1

Question: What is the probability of getting at least one head when two fair coins are tossed? Solution:

1. Identify the event: "at least one head". This is a perfect candidate for the complementary event rule.
2. The complementary event is "no heads", which means both coins are tails (TT).
3. Total possible outcomes when tossing two coins are HH, HT, TH, TT (4 outcomes).
4. Probability of "no heads" (TT) = 1/4.
5. Using the complementary rule: P(at least one head) = 1 - P(no heads) = 1 - 1/4 = 3/4.

Worked Example 2

Question: A single card is drawn from a standard deck of 52 cards. What is the probability that it is a King or a Spade? Solution:

1. Let Event A be drawing a King. P(A) = 4/52 (4 Kings in 52 cards).
2. Let Event B be drawing a Spade. P(B) = 13/52 (13 Spades in 52 cards).
3. These events are NOT mutually exclusive because there is one card that is both a King and a Spade (King of Spades).
4. Probability of drawing a King AND a Spade (P(A ∩ B)) = 1/52 (King of Spades).
5. Using the Addition Rule: P(A U B) = P(A) + P(B) - P(A ∩ B) P(King or Spade) = 4/52 + 13/52 - 1/52 = 17/52 - 1/52 = 16/52.
6. Simplify the fraction: 16/52 = 4/13.

Shortcuts & Tricks

• "At least one" = 1 - P(None): This is the most powerful shortcut for these types of questions. Always look for opportunities to use it. For example, probability of at least one success in N trials = 1 - (probability of all failures in N trials).
• Independent Events: If events are independent, P(A and B) = P(A) * P(B). Don't overcomplicate it with conditional probability.
• Mutually Exclusive Events: If events cannot happen together, P(A or B) = P(A) + P(B). The overlap P(A ∩ B) is zero.
• Visualize: For 'OR' problems, mentally (or quickly sketch) a Venn diagram to ensure you correctly identify and subtract the overlap.

Common Mistakes

1. Forgetting the Overlap: A common error in addition rule problems is to simply add P(A) + P(B) without subtracting P(A ∩ B) when events are not mutually exclusive.
2. Confusing Independence and Mutual Exclusivity: These are distinct concepts. Independent events don't affect each other; mutually exclusive events cannot occur simultaneously. An event cannot be both independent and mutually exclusive (unless one event has 0 probability).
3. Incorrectly Calculating P(None): For "at least one" problems, ensure P(none) is calculated correctly, especially when multiple trials are involved (e.g., P(none in 3 trials) = P(failure)^3 if independent).
4. Not Simplifying Fractions: Always simplify your final probability fractions to their lowest terms.

Core Formula

Probability is the measure of the likelihood of an event occurring. It is always a value between 0 and 1, inclusive.

P(E) = (Number of Favourable Outcomes) / (Total Number of Possible Outcomes)

Where:

• P(E) is the probability of event E.
• Favourable Outcomes are the outcomes where the event E occurs.
• Total Outcomes (also called the Sample Space) are all possible outcomes of an experiment.

Worked Example 1: Single Die Roll

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

Solution:

1. Identify Total Outcomes (Sample Space): When rolling a die, the possible outcomes are {1, 2, 3, 4, 5, 6}. So, Total Outcomes = 6.
2. Identify Favourable Outcomes: We want an even number, so the favourable outcomes are {2, 4, 6}. So, Favourable Outcomes = 3.
3. Apply the Formula: P(Even Number) = (Favourable Outcomes) / (Total Outcomes) = 3 / 6 = 1/2.

Worked Example 2: Two Coin Tosses

Question: What is the probability of getting exactly one head when two fair coins are tossed simultaneously?

Solution:

1. Identify Total Outcomes (Sample Space): When two coins are tossed, the possible outcomes are {HH, HT, TH, TT}. So, Total Outcomes = 4.
2. Identify Favourable Outcomes: We want exactly one head, so the favourable outcomes are {HT, TH}. So, Favourable Outcomes = 2.
3. Apply the Formula: P(Exactly one head) = (Favourable Outcomes) / (Total Outcomes) = 2 / 4 = 1/2.

Shortcuts & Tricks

1. Complement Rule: The probability of an event not happening is 1 minus the probability of it happening. P(E') = 1 - P(E). This is useful when calculating the probability of "at least one" or "not E".
   • Example: Probability of not getting a 6 on a die = 1 - P(getting a 6) = 1 - 1/6 = 5/6.
2. Counting Sample Space:
   • For 'n' coins: Total Outcomes = 2^n.
   • For 'n' dice: Total Outcomes = 6^n.
   • For 'n' cards drawn without replacement from 'N' cards: N * (N-1) * ... * (N-n+1) or N P n (Permutations).
3. Card Deck Basics: A standard deck has 52 cards: 4 suits (Hearts, Diamonds, Clubs, Spades), 13 cards per suit. Hearts and Diamonds are Red (26 cards); Clubs and Spades are Black (26 cards). Each suit has A, 2-10, J, Q, K. Face cards are J, Q, K (12 total).

Common Mistakes

1. Incorrectly Counting Sample Space: Forgetting outcomes (e.g., HT and TH are distinct for two coins) or double-counting.
2. Confusing "AND" with "OR": "AND" usually implies multiplication (for independent events), "OR" implies addition (for mutually exclusive events, or P(A)+P(B)-P(A and B) for general events).
3. Not Simplifying Fractions: Always simplify your probability fractions to their lowest terms.