Probability – Bag Contains Blue Balls, Red Balls, Blue Marbles, Yellow Marbles (Aptitude, Formulas, Examples, Answer, Explanation)

Formulas and Quick Tricks for Probability

1. Def. of Probability: Probability is the measure of uncertainty of any event (any phenomenon happened or bound to happen)
2. Experiment: Any phenomenon like rolling a dice, tossing a coin, drawing a card from a well-shuffled deck, etc.
3. Outcome: The Result of any event; like number appearing on a dice, side of a coin, drawn out card, etc.
4. Sample Space: The set of all possible outcomes.
5. Event: Any combination of possible outcomes or the subset of sample space; like getting an even number on rolled dice, getting a head/tail on a flipped coin, drawing out a king/queen/ace of any suit.
6. Probability = (Number of a Favourable outcome) / (Total number of outcomes)
   or, P = n (E) / n (S)
   where,
   P(A) is the probability of an event “A”
   n(E) is the number of favourable outcomes
   n(S) is the total number of events in the sample space
7. Odds in Favour of the Event: Odds in the favor of any event is the ratio of the number of ways that an outcome can occur to the number of ways it cannot occur.
8. Odds Against the Event: Odds against any event is the ratio of the number of ways that an outcome cannot occur to the number of ways it can occur.
9. Probability Range: 0 ≤ P(A) ≤ 1
10. Rule of Addition: P(A∪B) = P(A) + P(B) – P(A∩B)
11. Rule of Complementary Events: P(A’) + P(A) = 1
12. Disjoint Events: P(A∩B) = 0
13. Independent Events: P(A∩B) = P(A) . P(B)
14. Conditional Probability: P(A | B) = P(A∩B) / P(B)
15. Bayes Formula: P(A | B) = P(B | A) . P(A) / P(B)

Questions on Probability

Q.1: In a cycle race there are 5 persons named as J,K,L,M,N participated for 5 positions. How many number of ways can M finishes always before N?
A. 20
B. 36
C. 55
D. 60
Answer: D. 60
Explanation: Total number of ways in which 5 persons can finish is 5! = 120 (there are no ties)
Now, in half of these ways M can finish before N

Q.2: A box contains 3 blue marbles, 4 red, 6 green marbles and 2 yellow marbles. If three marbles are picked at random, what is the probability that they are all blue?
A. 1/455
B. 2/455
C. 4/455
D. 1/91
Answer: A. 1/455
Explanation: Given that there are three blue marbles, four red marbles, six green marbles and two yellow marbles.
Probability that all the three marbles picked at random are blue = 3C₃ / 15C₃ = (3 * 2 * 1) / (15 * 14 * 13) = 1/455

Probability – Basic Concepts, Bag and Ball Problems (Part 1) (Quantitative Aptitude made Simpler)

Q.3: A Bag contains 6 Blue Balls and 4 Red Balls. 4 balls are picked at random. What is the probability that 2 are Blue and 2 are Red?
A. 1/5
B. 1/7
C. 3/7
D. 3/8
Answer: C. 3/7
Explanation: Watch the Probability Series (Part I) Video

Q.4: A Bag contains 6 Blue Balls and 4 Red Balls. 3 balls are picked at random. What is the probability that 3 are Blue or 3 are Red?
A. 1/5
B. 1/6
C. 1/7
D. 1/8
Answer: A. 1/5
Explanation: Watch the Probability Series (Part I) Video

Probability – Basic Concepts, Bag and Ball Problems (Part 2) (Quantitative Aptitude made Simpler)

Q.5: A Bag contains 6 Blue Balls and 4 Red Balls. 5 balls are picked at random. What is the probability that 3 are Blue and 2 are Red or 2 are Blue and 3 are Red?
A. 3/7
B. 4/7
C. 5/7
D. 6/7
Answer: C. 5/7
Explanation: Watch the Probability Series (Part II) Video

Q.6: A Bag contains 6 Blue Balls and 4 Red Balls. 3 balls are picked at random. What is the prob. that none of them is Red?
A. 1/3
B. 1/5
C. 1/6
D. 1/7
Answer: C. 1/6
Explanation: Watch the Probability Series (Part II) Video


Q.7: The probability that A speaks truth is 3/5 and that of B speaking truth is 4/7. What is the probability that they agree in stating the same fact?
A. 12/35
B. 15/35
C. 18/35
D. 21/35
Answer: C. 18/35
Explanation: If both agree stating the same fact, either both of them speak truth or both speak false.
Probability = 3/5 * 4/7 + 2/5 * 3/7
= 12/35 + 6/35 = 18/35

Q.8: If a card is drawn from a well shuffled pack of cards, the probability of drawing a spade or a king is:
A. 19/52
B. 17/52
C. 4/13
D. 5/13
Answer: C. 4/13
Explanation: P(SuK) = P(S) + P(K) – P(SnK), where S denotes spade and K denotes king.
P(SuK) = 13/52 + 4/52 – 1/52 = 4/13


Q.9: Three 6 faced dice are thrown together. The probability that all the three show the same number on them is:
A. 1/64
B. 1/36
C. 5/9
D. 5/12
Answer: B. 1/36
Explanation: If all 3 numbers have to be same; basically we want triplets. 111, 222, 333, 444, 555 and 666. Those are six in number.
Further the three dice can fall in 6 * 6 * 6 = 216 ways
Hence the probability is 6/216 = 1/36

Q.10: A box contains 3 blue marbles, 4 red, 6 green marbles and 2 yellow marbles. If four marbles are picked at random, what is the probability that none is blue?
A. 17/91
B. 33/91
C. 51/91
D. 65/91
Answer: B. 33/91
Explanation: Given that there are 3 blue marbles, 4 red marbles, 6 green marbles and 2 yellow marbles.
When 4 marbles are picked at random, then the probability that none is blue is = 12C₄/15C₄
=>(12 * 11 * 10 * 9)/(15 * 14 * 13 * 12) = 33/91

Q.11: Out of 15 consecutive numbers, 2 are chosen at random. The probability that they are both odds or both primes is:
A. 13/21
B. 10/19
C. 11/15
D. Cannot be determined
Answer: D. Cannot be determined
Explanation: There is no definite formula for finding prime numbers among 15 consecutive numbers. Hence the probability cannot be determined.

Q.12: 10 books are placed at random in a shelf. The probability that a pair of books will always be together is:
A. 1/10
B. 1/5
C. 3/10
D. 9/10
Answer: B. 1/5
Explanation: 10 books can be rearranged in 10! ways
Consider the 2 books taken as a pair then, number of favourable ways of getting these two books together is 9! 2!
Required probability = 1/5

Q.13: The probability of a lottery ticket being a prized ticket is 0.2. When 4 tickets are purchased, the probability of winning a prize on atleast one ticket is:
A. 0.4869
B. 0.5834
C. 0.5904
D. 0.6234
Answer: C. 0.5904
Explanation: P(winning prize atleast on one ticket)
= 1 – P('Losing on all tickets')
= 1 – (0.8)^4 = (1 + (0.8)^2)(1 – (0.8)^2)
= (1.64)(0.36) = 0.5904

Q.14: Tickets numbered 1 to 20 are mixed up and then a ticket is drawn at random. What is the probability that the ticket drawn has a number which is a multiple of 3 or 5?
A. 1/2
B. 2/5
C. 8/15
D. 9/20
Answer: D. 9/20
Explanation: Here, S = {1, 2, 3, 4, ...., 19, 20}
Let E = event of getting a multiple of 3 or 5 = {3, 6 , 9, 12, 15, 18, 5, 10, 20}
P(E) = n(E)/n(S) = 9/20