Probability

Maths-11 Probability: Questions and Step-by-Step Solutions for all Competitive Exams

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Probability

Maths-11 Probability: Questions and Step-by-Step Solutions for all Competitive Exams

Let’s learn probability from the very beginning. Here are the Key formulas and step-by-step worked examples (digit-by-digit arithmetic).

What is probability?

Probability measures how likely an event is to happen.
If all outcomes are equally likely:

“Probability of event ” E=P(E)=”Number of favourable outcomes” /”Total number of possible outcomes”

Probability values lie between 0 and 1:

  • P (E) = 0 means impossible.
  • P (E) = 1means certain.
  • You can also write probability as a fraction, decimal, or percent.

Key words

  • Experiment: a trial (e.g., tossing a coin).
  • Outcome: result of one trial (e.g., head).
  • Sample space (S): list of all possible outcomes.
  • Event: a set of outcomes (e.g., “get an even number” when rolling a die).
  • Favourable outcomes: outcomes that match the event.

Examples of sample spaces

  • Toss 1 fair coin: S = ( H,T ). Total outcomes = 2.
  • Toss 2 fair coins: S = (HH, HT, TH, TT ) . Total outcomes = 4.
  • Roll a fair die: S = (1,2,3,4,5,6 ) . Total outcomes = 6.
  • Pick a card from a 52-card deck (standard): total outcomes = 52.

Simple worked examples

Example 1 — coin

Q: Toss a fair coin. What is the probability of getting a Head?

Work:

  1. Sample space S = (H T ). Total = 2.
  2. Favourable outcomes for Head = {H}. Count = 1.
  3. P (Head) = 1/2. As decimal = 0.5. As percent = 50 %.

Answer: 1/2 = 0.5 = 50%

Example 2 — die (single roll)

Q: Roll a fair die. What is the probability of getting an even number?

Work (digit-by-digit):

  1. Sample space S = (1,2,3,4,5,6) Total outcomes = 6.
  2. Even numbers in S = {2,4,6}. Count favourable = 3.
  3. Probability = 3/6 Compute stepwise:
    • 3/6 = 0.5
    • As percent: 0,5×100 = 50%

Answer: 3/6 = 1/2 = 50%

Example 3 — two coins

Q: Toss two fair coins. Probability of exactly one Head?

Work:

  1. Sample space = (HH,HT,TH,TT). Total = 4.
  2. Exactly one Head outcomes = {HT, TH}. Count = 2.
  3. Probability = 2/4 = 1/2= 0.5 = 50%.

Answer: 1/2

Example 4 — cards (basic)

Q: Pick one card from a standard 52-card deck. What is the probability it is a heart?

Work:

  1. Total outcomes = 52.
  2. Hearts in a deck = 13. (One suit has 13 cards.)
  3. Probability 13/52 . Compute: 13/52 = 0.25 . Percent: 0.25x 100.

Answer: 13/52= 1/4= 0.25= 25%.

Rules & shortcuts

A. Complement rule

If E is an event and E’ is its complement (event does not happen),

P (E’)= 1- P(E)

Useful when “at least one” or “not” is easier.

Example: Probability at least one head in two coin tosses = = 1- P (no head). ( No head = TT has probability 1/4 . So at least one head = 1- 1/4 = 3/4 .

B. Addition rule (mutually exclusive events)

If and cannot both happen together (mutually exclusive),

P (A U B)= P(A) + P(B)

Example: Roll a dice. Probability of (roll 1) or (roll 2) = 1/6 + 1/6 = 2/6 = 1/3.

C. General addition rule (overlap)

If not mutually exclusive,

P(AUB)= P(A) + P(B) – (A ∩ B)

Where P( A ∩ B) There is a probability that both happen.

Example: Draw a card. Probability(card is heart or queen) = P(heart) + P (queen) – P(heart and queen) = 13/52 + 4/52 – 1/52 = 16/52

D. Multiplication rule (independent events)

If events and are independent,

P (A ∩ B) = P(A) x P (B)

Example: Toss two fair coins. Probability both heads = P(H) x P(H) = (1/2) x (1/2) = 1/4

More worked examples (step-by-step with arithmetic)

Example 5 — complement

Q: Roll a dice. What is probability of not getting a 6?

Work:

  1. P (6) = 1/6
  2. Complement: P(not 6) + 1-1/6 = (6/6 -1/6) = 5/6.
  3. Decimal: 5/6 = 0.833333. Percent ≈ 83.33%.

Answer: 83.33%.

Example 6 — “at least one”

Q: Toss two coins. Probability of at least one Head?

Work:

  1. Easier via complement: no head = TT has probability 1/4.
    • Reason: TT is 1 outcome of 4 equally likely outcomes → 1/4.
  2. So P (at least one head) = 1-1/4 = 3/4. Decimal = 0.75. Percent 75%.

Answer: 3/4 = 0.75 = 75%.

Example 7 — dice sum (two dice)

Q: Roll two fair dice. What is probability the sum is 7?

Work:

  1. Sample space total outcomes = 6×6 = 36 (ordered pairs (1,1) … (6,6)).
  2. Pairs giving sum 7: (1,6),(2,5),(3,4),(4,3),(5,2),(6,1). Count = 6.
  3. P = 6/36 = 1/6 Decimal = 0.166666. Percent ≈ 16.67%.

Answer: 16.67 %

Example 8 — card overlap

Q: Draw one card. Probability it is a red Queen?

Work:

  1. Red suits: hearts and diamonds. Queens: one in each suit. Red queens = queen of hearts + queen of diamonds ⇒ 2 cards.
  2. Total = 52. P = 2/52 = 1/26 . Decimal = 0.03846 . Percent ≈ 3.846%.

Answer: 3.846%

Example 9 — dependent event (without replacement)

Q: From a bag with 3 red and 2 blue balls, pick two balls without replacement. What is probability both are red?

Work (step-by-step):

  1. First draw: probability red = 3/(3+2) = 3/5.
  2. After one red is removed, remaining red = 2, total remaining = 4. Probability second red = 2/4=1/2
  3. Multiply (dependent sequence): P (red then red )= (3/5) x (1/2). Compute:
    • 3 x 1 = 3
    • Denominator 5
    • So 3/10 = 0.3 = 30% .

Answer: 30 %

Example 10 — independent events (with replacement)

Q: Same bag (3 red, 2 blue). Pick two balls with replacement. Probability both red?

Work:

  1. With replacement probabilities stay same. P(red first) = 3/5. P(red second) = 3/5.
  2. Multiply: (3/5) x (3/5) = 9/25. Decimal = 0.36 . Percent 36% .

Answer: 36%

Visualizing probability (quick intuition)

  • Closer the probability is to 1 → more likely.
  • 0.5 (50%) means equally likely to happen or not.
  • Use dice/cards/coins to practice counting outcomes.

Short practice set (10 questions) — try first, then check answers

  1. Toss a fair coin once. P(head) =
  2. Roll a die. P(odd number) =
  3. Draw a card from 52. P(ace) =
  4. Roll two dice. P(sum = 11) =
  5. From digits 0–9, pick one at random. P(≤3) =
  6. Toss two coins. P(both tails) =
  7. From a bag with 4 white and 6 black balls, pick one; P(white) =
  8. Same bag, pick two without replacement; P(both white) =
  9. From 52 cards, P(a spade or a king) =
  10. Toss 3 fair coins. P(exactly two heads) =

Answers (brief, with one-line work)

  1. Coin:1/2 = 50 %.
  2. Die odd numbers {1,3,5} → 3/6 = 1/2 = 50% .
  3. Aces = 4 → 4/52 = 1/13 = 7.692 %
  4. Two dice sum 11 pairs: (5,6),(6,5) → 2/36 = 1/18 ≈ 5.56%.
  5. Digits 0–9 total 10; ≤3 are {0,1,2,3} count 4 → 4/10 = 2/5 = 0.4 = 40%.
  6. Two tails TT out of 4 → 1/4 = 25 %
  7. 4 white out of 10 → 4/10 = 2/5 = 25%
  8. Without replacement: (4/10)×(3/9) = 12/90 = 2/15 ≈ 13.33%.
  9. P(spade) = 13/52 = 1/4. P(king) = 4/52 = 1/13. Overlap king of spades = 1/52. So total = 13/52 + 4/52 − 1/52 = 16/52 = 4/13 ≈ 30.77%.
  10. Toss 3 coins: sample size 8. Exactly two heads outcomes {HHT, HTH, THH} count 3 → 3/8 = 37.5%.

Final tips & exam tricks

  • Count carefully: list outcomes if small sample space (dice, coins).
  • Use the complement method for “at least one” and similar questions.
  • For cards/draws: pay attention to replacement vs no replacement.
  • Convert probability to decimal or percent if asked.
  • Simplify fractions before converting for clean answers.

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