Statistics Probability
Quick Theory Refresher
Probability measures how likely an event is to occur. For equally-likely outcomes,
P(E) = (Number of favourable outcomes) / (Total possible outcomes).
The value always lies between 0 (impossible) and 1 (certain). The complement rule
P(not E) = 1 – P(E) helps when direct counting is messy.
Two events are mutually exclusive if both cannot happen together (P(A∩B)=0) and independent if one doesn’t affect the other (P(A∩B)=P(A)·P(B)).
For compound events:
- Addition: P(A∪B) = P(A)+P(B)–P(A∩B)
- Conditional: P(A|B) = P(A∩B) / P(B)
Mean, median & mode describe central tendency, while variance/standard-deviation quantify spread.
For grouped data, mean = Σ(fx)/Σf and variance = [Σf(x–mean)²]/Σf.
Remember: in a normal distribution, ≈68 % values lie within ±1σ and ≈95 % within ±2σ—handy for quick elimination in range-based questions.
Practice MCQs
-
Easy – A fair die is rolled once. The probability of getting a prime number is
A) 1/6
B) 1/3
C) 1/2
D) 2/3Answer
Correct: C. Primes on a die = {2,3,5}; 3/6 = 1/2. -
Easy – Which of the following cannot be a probability value?
A) 0
B) 0.5
C) 1
D) 1.2Answer
Correct: D. Probability must lie in [0,1]. -
Easy – The median of 7,4,9,7,5,8,6 is
A) 5
B) 6
C) 7
D) 8Answer
Correct: C. Ordered data 4,5,6,7,7,8,9 → middle value 7. -
Easy – Two coins are tossed simultaneously. Probability of exactly one head is
A) 1/4
B) 1/2
C) 3/4
D) 1Answer
Correct: B. Favourable = {HT,TH}; 2/4 = 1/2. -
Easy – If P(A)=0.3, then P(not A) equals
A) 0.3
B) 0.7
C) 1
D) 0Answer
Correct: B. Complement rule 1–0.3 = 0.7. -
Easy – The mode of 2,3,3,5,5,5,7 is
A) 2
B) 3
C) 5
D) 7Answer
Correct: C. 5 appears most frequently. -
Medium – A card is drawn from a 52-card deck. Probability that it is a king or a heart is
A) 16/52
B) 17/52
C) 1/13
D) 4/13Answer
Correct: A. P(King)+P(Heart)–P(King of Hearts)=4/52+13/52–1/52=16/52. -
Medium – Two dice are rolled. Probability that the sum is 9 is
A) 1/9
B) 1/12
C) 1/6
D) 4/36Answer
Correct: D. Favourable pairs (3,6),(4,5),(5,4),(6,3) → 4/36. -
Medium – Mean of first 5 natural numbers is
A) 3
B) 3.5
C) 4
D) 5Answer
Correct: A. (1+2+3+4+5)/5 = 15/5 = 3. -
Medium – If events A and B are independent with P(A)=0.4, P(B)=0.5, then P(A∩B)=
A) 0.2
B) 0.4
C) 0.5
D) 0.9Answer
Correct: A. 0.4×0.5 = 0.2. -
Medium – A bag has 4 red, 3 blue, 5 green balls. One ball is drawn at random. Probability it is not green is
A) 5/12
B) 7/12
C) 1/2
D) 3/4Answer
Correct: B. (4+3)/12 = 7/12. -
Medium – The variance of 2,4,6,8,10 is
A) 6
B) 8
C) 10
D) 12Answer
Correct: B. Mean=6; Σ(x–6)²/5 = (16+4+0+4+16)/5 = 40/5 = 8. -
Medium – In a class of 40, 25 passed in maths, 20 in science, 5 failed in both. Number passing both subjects is
A) 10
B) 15
C) 20
D) 25Answer
Correct: A. n(M∪S)=35; 25+20–x=35 → x=10. -
Hard – Three coins are tossed. Probability of at least two tails is
A) 1/2
B) 3/8
C) 1/4
D) 5/8Answer
Correct: A. Favourable {HTT,THT,TTH,TTT} = 4/8 = 1/2. -
Hard – A box has 2 white, 3 black, 4 red balls. Two balls are drawn without replacement. Probability both are black is
A) 1/15
B) 3/36
C) 1/12
D) 2/9Answer
Correct: C. (3/9)×(2/8) = 6/72 = 1/12. -
Hard – The standard deviation of the data 5,5,5,5,5 is
A) 0
B) 1
C) 5
D) 25Answer
Correct: A. No dispersion; SD=0. -
Hard – A single letter is chosen at random from the word “PROBABILITY”. Probability it is a vowel is
A) 4/11
B) 3/11
C) 1/3
D) 2/11Answer
Correct: A. Vowels O,A,I,I → 4 out of 11 letters. -
Hard – Two cards are drawn with replacement from a 52-card deck. Probability both are aces is
A) 1/221
B) 1/169
C) 1/2704
D) 4/52Answer
Correct: B. (4/52)×(4/52) = 16/2704 = 1/169. -
Hard – Mean of 10 items is 15. If one item 25 is replaced by 35, the new mean is
A) 15
B) 16
C) 17
D) 18Answer
Correct: B. Increase in total = 10; new mean = 15 + 10/10 = 16. -
Hard – A die is rolled twice. Probability that the second roll is strictly greater than the first is
A) 5/12
B) 1/2
C) 7/12
D) 15/36Answer
Correct: A. Count 15 favourable ordered pairs → 15/36 = 5/12. -
Hard – In a normal distribution, approximately what percentage of values lie within μ±1.5σ?
A) 68 %
B) 86.6 %
C) 95 %
D) 99.7 %Answer
Correct: B. Empirical rule gives ≈86.6 %. -
Hard – A bag contains 5 defective and 15 non-defective bulbs. Two bulbs are drawn at random without replacement. Probability that exactly one is defective is
A) 15/76
B) 35/76
C) 50/76
D) 15/38Answer
Correct: B. (5C1×15C1)/(20C2) = 75/190 = 35/76. -
Hard – If the probability that A solves a problem is 3/5 and for B it is 2/3, and they try independently, the probability that the problem is solved by at least one of them is
A) 2/5
B) 13/15
C) 3/5
D) 11/15Answer
Correct: B. 1–(1–3/5)(1–2/3)=1–(2/5)(1/3)=1–2/15=13/15. -
Hard – The mean of first n natural numbers is 15. Then n equals
A) 15
B) 29
C) 30
D) 31Answer
Correct: B. n(n+1)/2n = (n+1)/2 = 15 → n=29. -
Hard – A railway time-table shows 90 % trains arrive on time. If 5 trains are chosen at random, the probability that at least one is late is
A) (0.9)^5
B) 1–(0.1)^5
C) 1–(0.9)^5
D) (0.1)^5Answer
Correct: C. Complement of all on-time = 1–(0.9)^5.
Shortcuts & Tips
- Complement trick: “At least one” questions → find 1 – P(none).
- Dice sums: Count pairs symmetrically; 7 has max combos (6).
- Card shortcuts: 13 cards per suit; 4 each of kings, queens, etc.
- Mean shift: If every observation increases by k, mean increases by k; variance unchanged.
- Empirical rule: 68-95-99.7 for normal curve—use to eliminate wild options.
- “Without replacement” → denominator drops by 1 for second draw.
- Mode in grouped data: use formula Mode = L + [(f1–f0)/(2f1–f0–f2)]×h only if boundary values are close; else inspect highest frequency class.
- Time-saver: Reduce fractions before multiplying to keep numbers small.
- Double-check: Always verify whether events are mutually exclusive or independent—don’t mix addition & multiplication rules.
- Rough column: While calculating SD, write (x–mean)² in a side column; halves arithmetic errors.
BETA
