Chapter 14: Probability

Question 1: The probability of an event that is certain to happen is:
(A) 0

(B) 1

(C) -1

(D) 0.5

Solution: (B) 1
Reason: The probability of a sure (certain) event is always 1.
Question 2: Which of the following cannot be the probability of an event?
(A) 2/3

(B) -1.5

(C) 15%

(D) 0.7

Solution: (B) -1.5
Reason: Probability of an event always lies between 0 and 1 (inclusive). It cannot be negative.
Question 3: If $P(E) = 0.05$, then the probability of 'not E' is:
(A) 0.05

(B) 0.5

(C) 0.95

(D) 0.9

Solution: (C) 0.95
Step 1: $P(\text{not } E) = 1 - P(E)$.
Step 2: $1 - 0.05 = 0.95$.
Question 4: A die is thrown once. The probability of getting a prime number is:
(A) 2/3

(B) 1/3

(C) 1/2

(D) 1/6

Solution: (C) 1/2
Step 1: Total outcomes = {1, 2, 3, 4, 5, 6} (6 outcomes).
Step 2: Prime numbers = {2, 3, 5} (3 outcomes).
Step 3: Probability $= 3/6 = 1/2$.
Question 5: A card is drawn from a well-shuffled deck of 52 cards. The probability that the card will not be an ace is:
(A) 1/13

(B) 4/13

(C) 12/13

(D) 3/13

Solution: (C) 12/13
Step 1: Number of aces = 4. Probability of ace $= 4/52 = 1/13$.
Step 2: Probability of not ace $= 1 - 1/13 = 12/13$.
Question 6: Two coins are tossed simultaneously. The probability of getting at most one head is:
(A) 1/4

(B) 1/2

(C) 3/4

(D) 1

Solution: (C) 3/4
Step 1: Outcomes: {HH, HT, TH, TT}. Total = 4.
Step 2: At most one head means 0 or 1 head: {HT, TH, TT}. Favourable = 3.
Step 3: Probability $= 3/4$.
Question 7: A bag contains 5 red balls and some blue balls. If the probability of drawing a blue ball is double that of a red ball, then the number of blue balls in the bag is:
(A) 5

(B) 10

(C) 15

(D) 20

Solution: (B) 10
Step 1: Let blue balls $= x$. Total $= 5+x$.
Step 2: $P(B) = \frac{x}{5+x}$, $P(R) = \frac{5}{5+x}$.
Step 3: Given $P(B) = 2 P(R) \Rightarrow \frac{x}{5+x} = 2(\frac{5}{5+x}) \Rightarrow x = 10$.
Question 8: The probability of selecting a rotten apple randomly from a heap of 900 apples is 0.18. What is the number of rotten apples in the heap?
(A) 162

(B) 164

(C) 160

(D) 168

Solution: (A) 162
Step 1: $P(\text{Rotten}) = \frac{\text{Number of rotten}}{\text{Total}}$.
Step 2: $0.18 = \frac{N}{900} \Rightarrow N = 0.18 \times 900 = 162$.
Question 9: A game consists of tossing a one rupee coin 3 times and noting its outcome each time. Hanif wins if all the tosses give the same result i.e., three heads or three tails, and loses otherwise. Calculate the probability that Hanif will lose the game.
(A) 3/4

(B) 1/4

(C) 5/8

(D) 3/8

Solution: (A) 3/4
Step 1: Total outcomes = 8 (HHH, HHT, HTH, HTT, THH, THT, TTH, TTT).
Step 2: Winning outcomes (same result): {HHH, TTT} = 2.
Step 3: Losing outcomes = $8 - 2 = 6$. Probability $= 6/8 = 3/4$.
Question 10: A number is selected at random from the numbers 1 to 30. The probability that it is a prime number is:
(A) 1/3

(B) 2/3

(C) 1/6

(D) 11/30

Solution: (A) 1/3
Step 1: Primes between 1 and 30: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.
Step 2: Count = 10. Total numbers = 30.
Step 3: Probability $= 10/30 = 1/3$.

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