Friday, November 22, 2024

MCQ Mathematics Chapter 14: PROBABILITY, HS 1st year

 

MCQ 1: Sample Space and Events

If a coin is tossed twice, what is the sample space (S) for this experiment?

  1. {HH,HT,TH,TT}\{HH, HT, TH, TT\}
  2. {H,T}\{H, T\}
  3. {H,T,HH,TT}\{H, T, HH, TT\}
  4. {HHT,TTT}\{HHT, TTT\}

Answer: 1. {HH,HT,TH,TT}

MCQ 2: Types of Events

In the experiment of rolling a die, which event is considered a simple event?

  1. Getting an odd number
  2. Rolling a prime number
  3. Rolling a 3
  4. Rolling a number greater than 2

Answer: 3. Rolling a 3

MCQ 3: Complementary Events

The complementary event of rolling an even number on a die is:

  1. Rolling an odd number
  2. Rolling a number greater than 3
  3. Rolling a prime number
  4. Rolling a multiple of 2

Answer: 1. Rolling an odd number

MCQ 4: Probability Axioms

According to the axiomatic approach to probability, the probability of the sample space (S) is:

  1. 0
  2. 0.5
  3. 1
  4. Undefined

Answer: 3. 1

MCQ 5: Mutually Exclusive Events

Two events A and B are said to be mutually exclusive if:

  1. A∩B=ϕA \cap B = \phi
  2. A∩B=SA \cap B = S
  3. A∪B=SA \cup B = S
  4. A−B=ϕA - B = \phi

Answer: 1. A∩B=ϕA \cap B = \phi

MCQ 6: Probability Calculation

A die is rolled. What is the probability of getting a prime number?

  1. 12\frac{1}{2}
  2. 23\frac{2}{3}
  3. 13\frac{1}{3}
  4. 16\frac{1}{6}

Answer: 1. 12\frac{1}{2}
(Prime numbers on a die: 2, 3, 5; Total outcomes: 6; Probability = 36=12\frac{3}{6} = \frac{1}{2}.)

MCQ 7: Exhaustive Events

In the experiment of rolling a die, the events: A: 'number less than 4', B: 'number between 3 and 5', and C: 'number greater than 4' are:

  1. Mutually exclusive but not exhaustive
  2. Mutually exclusive and exhaustive
  3. Not mutually exclusive but exhaustive
  4. Neither mutually exclusive nor exhaustive

Answer: 2. Mutually exclusive and exhaustive

MCQ 8: Impossible Events

Which of the following represents an impossible event when rolling a die?

  1. Rolling a number less than 7
  2. Rolling a number greater than 6
  3. Rolling a number equal to 4
  4. Rolling an odd number

Answer: 2. Rolling a number greater than 6

MCQ 9: Compound Events

Which of the following is a compound event in the experiment of tossing three coins?

  1. At least one head appears
  2. Exactly one head appears
  3. At most one head appears
  4. All of the above

Answer: 4. All of the above

MCQ 10: Mutually Exclusive and Exhaustive Events

In an experiment where a die is rolled, which of the following sets of events is both mutually exclusive and exhaustive?

  1. A: 'odd number appears', B: 'even number appears'
  2. A: 'number greater than 3', B: 'number less than 4'
  3. A: 'prime number appears', B: 'number less than 5'
  4. None of the above

Answer: 1. A: 'odd number appears', B: 'even number appears'

MCQ 11: Event Algebra

If AA and BB are events, the set A∪BA \cup B represents:

  1. Elements that are in AA but not in BB
  2. Elements that are in BB but not in AA
  3. Elements that are in either AA, BB, or both
  4. Elements that are in both AA and BB

Answer: 3. Elements that are in either AA, BB, or both

MCQ 12: Probability of Event 'Not A'

If P(A)=25P(A) = \frac{2}{5}, what is P(not A)P(\text{not A})?

  1. 35\frac{3}{5}
  2. 25\frac{2}{5}
  3. 52\frac{5}{2}
  4. 12\frac{1}{2}

Answer: 1. 35\frac{3}{5}
(P(not A)=1−P(A)P(\text{not A}) = 1 - P(A))

MCQ 13: Probability Calculation

A bag contains 4 red discs, 3 blue discs, and 2 yellow discs. A disc is chosen at random. What is the probability of selecting a blue disc?

  1. 19\frac{1}{9}
  2. 13\frac{1}{3}
  3. 39\frac{3}{9}
  4. 12\frac{1}{2}

Answer: 3. 39=13\frac{3}{9} = \frac{1}{3}

MCQ 14: Valid Probability Assignments

Which of the following is NOT a valid probability assignment for outcomes in a sample space SS?

  1. 0.3,0.3,0.40.3, 0.3, 0.4
  2. 0.1,0.1,0.2,0.60.1, 0.1, 0.2, 0.6
  3. −0.1,0.5,0.6-0.1, 0.5, 0.6
  4. 0.25,0.25,0.25,0.250.25, 0.25, 0.25, 0.25

Answer: 3. −0.1,0.5,0.6-0.1, 0.5, 0.6
(Probabilities cannot be negative.)

MCQ 15: Events 'A and B'

If P(A)=0.5P(A) = 0.5, P(B)=0.4P(B) = 0.4, and P(A∩B)=0.2P(A \cap B) = 0.2, what is P(A∪B)P(A \cup B)?

  1. 0.70.7
  2. 0.90.9
  3. 0.60.6
  4. 1.01.0

Answer: 2. 0.90.9
(P(A∪B)=P(A)+P(B)−P(A∩B)=0.5+0.4−0.2=0.9P(A \cup B) = P(A) + P(B) - P(A \cap B) = 0.5 + 0.4 - 0.2 = 0.9)

MCQ 16: Exhaustive Events

Consider an experiment of rolling a die. Which of the following events are exhaustive?

  1. A: 'number less than 4', B: 'number greater than 4'
  2. A: 'even number appears', B: 'odd number appears'
  3. A: 'prime number appears', B: 'number less than 3'
  4. None of the above

Answer: 2. A: 'even number appears', B: 'odd number appears'

MCQ 17: Probability of Drawing Cards

What is the probability of drawing a black card from a standard deck of 52 cards?

  1. 1352\frac{13}{52}
  2. 2652\frac{26}{52}
  3. 14\frac{1}{4}
  4. 34\frac{3}{4}

Answer: 2. 2652=12\frac{26}{52} = \frac{1}{2}

MCQ 18: Tossing Coins

Three coins are tossed. What is the probability of getting exactly two heads?

  1. 14\frac{1}{4}
  2. 38\frac{3}{8}
  3. 12\frac{1}{2}
  4. 18\frac{1}{8}

Answer: 2. 38\frac{3}{8}
(Possible outcomes: HHT, HTH, THH out of 8 total outcomes.)

MCQ 19: Equally Likely Outcomes

If a sample space SS contains nn equally likely outcomes, the probability of any single outcome is:

  1. n1\frac{n}{1}
  2. n2n^2
  3. 1n\frac{1}{n}
  4. n!n!

Answer: 3. 1n\frac{1}{n}

MCQ 20: Probability of ‘A or B’

If P(A)=0.4P(A) = 0.4, P(B)=0.3P(B) = 0.3, and P(A∩B)=0.1P(A \cap B) = 0.1, then P(A∪B)P(A \cup B) is:

  1. 0.60.6
  2. 0.70.7
  3. 0.50.5
  4. 0.80.8

Answer: 4. 0.80.8
(P(A∪B)=P(A)+P(B)−P(A∩B)=0.4+0.3−0.1=0.8P(A \cup B) = P(A) + P(B) - P(A \cap B) = 0.4 + 0.3 - 0.1 = 0.8.)

MCQ 21: Complementary Events

If P(A)=0.7P(A) = 0.7, what is P(not A)P(\text{not A})?

  1. 0.30.3
  2. 0.50.5
  3. 1.71.7
  4. 0.20.2

Answer: 1. 0.30.3
(P(not A)=1−P(A)=1−0.7=0.3P(\text{not A}) = 1 - P(A) = 1 - 0.7 = 0.3.)

MCQ 22: Impossible Event

Which of the following probabilities corresponds to an impossible event?

  1. 00
  2. 0.50.5
  3. 11
  4. 1.51.5

Answer: 1. 00

MCQ 23: Mutually Exclusive Events

Two events are mutually exclusive if:

  1. They always occur together
  2. They cannot occur at the same time
  3. One event is the subset of the other
  4. They have at least one common element

Answer: 2. They cannot occur at the same time

MCQ 24: Tossing Three Coins

Three coins are tossed. What is the probability of getting at least one tail?

  1. 18\frac{1}{8}
  2. 34\frac{3}{4}
  3. 78\frac{7}{8}
  4. 58\frac{5}{8}

Answer: 3. 78\frac{7}{8}
(P(at least one tail)=1−P(no tails)=1−18=78P(\text{at least one tail}) = 1 - P(\text{no tails}) = 1 - \frac{1}{8} = \frac{7}{8}.)

MCQ 25: Rolling a Die

When a die is rolled, what is the probability of getting a number greater than 4?

  1. 23\frac{2}{3}
  2. 13\frac{1}{3}
  3. 16\frac{1}{6}
  4. 12\frac{1}{2}

Answer: 2. 13\frac{1}{3}
(Numbers greater than 4: 5, 6; Total outcomes: 6; Probability = 26=13\frac{2}{6} = \frac{1}{3}.)

MCQ 26: Probability of Compound Events

In an experiment of rolling two dice, what is the probability of getting a sum of 7?

  1. 16\frac{1}{6}
  2. 136\frac{1}{36}
  3. 536\frac{5}{36}
  4. 636\frac{6}{36}

Answer: 4. 636=16\frac{6}{36} = \frac{1}{6}
(Sum of 7 can occur as (1,6),(2,5),(3,4),(4,3),(5,2),(6,1)(1,6), (2,5), (3,4), (4,3), (5,2), (6,1): 6 outcomes out of 36.)

MCQ 27: Deck of Cards

What is the probability of drawing an ace from a standard deck of 52 cards?

  1. 113\frac{1}{13}
  2. 14\frac{1}{4}
  3. 152\frac{1}{52}
  4. 413\frac{4}{13}

Answer: 1. 113\frac{1}{13}

MCQ 28: Red and Blue Discs

A bag contains 6 red discs and 4 blue discs. If one disc is drawn at random, what is the probability that it is blue?

  1. 14\frac{1}{4}
  2. 25\frac{2}{5}
  3. 35\frac{3}{5}
  4. 46\frac{4}{6}

Answer: 2. 25\frac{2}{5}
(Probability = number of blue discstotal discs=410=25\frac{\text{number of blue discs}}{\text{total discs}} = \frac{4}{10} = \frac{2}{5}.)

MCQ 29: Probability of A and B

If P(A)=0.6P(A) = 0.6, P(B)=0.5P(B) = 0.5, and P(A∪B)=0.8P(A \cup B) = 0.8, then P(A∩B)P(A \cap B) is:

  1. 0.30.3
  2. 0.10.1
  3. 0.50.5
  4. 0.80.8

Answer: 2. 0.10.1
(P(A∩B)=P(A)+P(B)−P(A∪B)=0.6+0.5−0.8=0.1P(A \cap B) = P(A) + P(B) - P(A \cup B) = 0.6 + 0.5 - 0.8 = 0.1.)

MCQ 30: Drawing Cards

What is the probability of drawing a red king from a standard deck of 52 cards?

  1. 152\frac{1}{52}
  2. 126\frac{1}{26}
  3. 113\frac{1}{13}
  4. 252\frac{2}{52}

Answer: 2. 126\frac{1}{26}
(There are 2 red kings in a deck of 52 cards; Probability = 252=126\frac{2}{52} = \frac{1}{26}.)

MCQ 31: Event ‘A or B’

If P(A)=0.5P(A) = 0.5, P(B)=0.6P(B) = 0.6, and P(A∩B)=0.2P(A \cap B) = 0.2, what is P(A∪B)P(A \cup B)?

  1. 0.80.8
  2. 0.90.9
  3. 1.01.0
  4. 0.70.7

Answer: 2. 0.90.9
(P(A∪B)=P(A)+P(B)−P(A∩B)=0.5+0.6−0.2=0.9P(A \cup B) = P(A) + P(B) - P(A \cap B) = 0.5 + 0.6 - 0.2 = 0.9.)

MCQ 32: Probability of 'At Least One'

Two dice are rolled. What is the probability that at least one die shows a 6?

  1. 136\frac{1}{36}
  2. 1136\frac{11}{36}
  3. 16\frac{1}{6}
  4. 1136\frac{11}{36}

Answer: 4. 1136\frac{11}{36}
(The probability of at least one 6 = 1−P(no 6s)=1−2536=11361 - P(\text{no 6s}) = 1 - \frac{25}{36} = \frac{11}{36}.)

MCQ 33: Exhaustive Events

In the experiment of flipping three coins, which of the following sets of events is exhaustive?

  1. A: 'two heads appear', B: 'no tails appear'
  2. A: 'at least one tail appears', B: 'no heads appear'
  3. A: 'at least one head appears', B: 'at most two tails appear'
  4. A: 'no heads appear', B: 'at least two heads appear', C: 'exactly one head appears'

Answer: 4. A: 'no heads appear', B: 'at least two heads appear', C: 'exactly one head appears'

MCQ 34: Coin Toss

A fair coin is tossed three times. What is the probability of getting no heads?

  1. 12\frac{1}{2}
  2. 18\frac{1}{8}
  3. 38\frac{3}{8}
  4. 14\frac{1}{4}

Answer: 2. 18\frac{1}{8}
(Only one outcome, TTTTTT, has no heads; P=18P = \frac{1}{8}.)

MCQ 35: Mutually Exclusive and Exhaustive

In an experiment of rolling a die, which of the following represents mutually exclusive and exhaustive events?

  1. A: 'even number appears', B: 'prime number appears'
  2. A: 'even number appears', B: 'odd number appears'
  3. A: 'number less than 4', B: 'number greater than 4'
  4. None of these

Answer: 2. A: 'even number appears', B: 'odd number appears'

MCQ 36: Drawing Cards

What is the probability of drawing a face card from a deck of 52 cards?

  1. 313\frac{3}{13}
  2. 14\frac{1}{4}
  3. 413\frac{4}{13}
  4. 913\frac{9}{13}

Answer: 1. 313\frac{3}{13}
(12 face cards in total; Probability = 1252=313\frac{12}{52} = \frac{3}{13}.)

MCQ 37: Mutually Exclusive Events

If events A and B are mutually exclusive, then:

  1. P(A∩B)=0P(A \cap B) = 0
  2. P(A∪B)=1P(A \cup B) = 1
  3. P(A)+P(B)=1P(A) + P(B) = 1
  4. P(A)×P(B)=0P(A) \times P(B) = 0

Answer: 1. P(A∩B)=0P(A \cap B) = 0

MCQ 38: Probability of Compound Events

Two dice are rolled. What is the probability of getting a sum of 8?

  1. 536\frac{5}{36}
  2. 16\frac{1}{6}
  3. 736\frac{7}{36}
  4. 13\frac{1}{3}

Answer: 1. 536\frac{5}{36}
(Sum of 8: (2,6),(3,5),(4,4),(5,3),(6,2)(2,6), (3,5), (4,4), (5,3), (6,2).)

MCQ 39: Random Events

In the experiment of rolling a die twice, what is the probability of getting at least one 4?

  1. 1136\frac{11}{36}
  2. 12\frac{1}{2}
  3. 16\frac{1}{6}
  4. 1336\frac{13}{36}

Answer: 1. 1136\frac{11}{36}
(1−P(no 4s)=1−2536=11361 - P(\text{no 4s}) = 1 - \frac{25}{36} = \frac{11}{36}.)

MCQ 40: Deck of Cards

What is the probability of drawing a queen of hearts from a standard deck of 52 cards?

  1. 113\frac{1}{13}
  2. 152\frac{1}{52}
  3. 452\frac{4}{52}
  4. 213\frac{2}{13}

Answer: 2. 152\frac{1}{52}

MCQ 41: Drawing Marbles

A bag contains 5 red marbles, 7 blue marbles, and 3 green marbles. What is the probability of drawing a green marble?

  1. 15\frac{1}{5}
  2. 315\frac{3}{15}
  3. 13\frac{1}{3}
  4. 14\frac{1}{4}

Answer: 2. 315=15\frac{3}{15} = \frac{1}{5}

MCQ 42: Dice Experiment

If two dice are thrown, what is the probability of getting doubles (both dice showing the same number)?

  1. 112\frac{1}{12}
  2. 16\frac{1}{6}
  3. 13\frac{1}{3}
  4. 536\frac{5}{36}

Answer: 2. 16\frac{1}{6}
(Doubles: (1,1),(2,2),(3,3),(4,4),(5,5),(6,6)(1,1), (2,2), (3,3), (4,4), (5,5), (6,6); Total = 6/36 = 1/6.)

MCQ 43: Complementary Events

If P(A)=0.35P(A) = 0.35, what is P(not A)P(\text{not A})?

  1. 0.550.55
  2. 0.750.75
  3. 0.650.65
  4. 0.850.85

Answer: 3. 0.650.65
(P(not A)=1−P(A)=1−0.35=0.65P(\text{not A}) = 1 - P(A) = 1 - 0.35 = 0.65.)

MCQ 44: Axioms of Probability

Which of the following statements violates the axioms of probability?

  1. P(A)=0.3P(A) = 0.3
  2. P(A)=−0.5P(A) = -0.5
  3. P(A)=1P(A) = 1
  4. P(A)=0P(A) = 0

Answer: 2. P(A)=−0.5P(A) = -0.5

MCQ 45: Rolling Dice

What is the probability of rolling a number less than 3 on a standard die?

  1. 12\frac{1}{2}
  2. 13\frac{1}{3}
  3. 16\frac{1}{6}
  4. 23\frac{2}{3}

Answer: 2. 13\frac{1}{3}
(Numbers less than 3: 1,21, 2; Probability = 26=13\frac{2}{6} = \frac{1}{3}.)


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