Probability is a fundamental concept in mathematics and statistics that quantifies the likelihood of an event occurring. It is used to model and predict uncertain outcomes in various fields such as physics, finance, biology, and social sciences. The foundation of probability lies in the understanding of random experiments, where the outcome is not known with certainty, but follows a set of possible outcomes.
Probability MCQs with answers practice now
Key aspects of probability:
- Probability Space: A probability space consists of three elements – a sample space (S), which is the set of all possible outcomes of the experiment; an event space (E), which is a set of specific outcomes from the sample space; and a probability measure (P), which assigns a non-negative real number between 0 and 1 to each event, representing the likelihood of that event occurring.
- Probability of an Event: The probability of an event A, denoted by P(A), represents the likelihood that event A will occur. It is calculated by dividing the number of favorable outcomes to A by the total number of outcomes in the sample space.
- Basic Rules of Probability: Probability is subject to certain fundamental rules, such as the sum rule (P(A ∪ B) = P(A) + P(B) – P(A ∩ B)) and the product rule (P(A ∩ B) = P(A) * P(B|A)), where A and B are events.
- Conditional Probability: Conditional probability measures the likelihood of an event occurring given that another event has already occurred. It is denoted by P(A|B) and is calculated as the probability of the joint event A and B divided by the probability of event B (if P(B) > 0).
- Independence: Two events A and B are independent if the occurrence of one does not affect the occurrence of the other. Mathematically, this means P(A ∩ B) = P(A) * P(B).
- Bayes’ Theorem: Bayes’ theorem is a fundamental theorem in probability that allows us to update our beliefs about an event in the light of new evidence. It relates the conditional probabilities of two events in opposite orders: P(A|B) = P(B|A) * P(A) / P(B).
Probability theory has practical applications in various fields, including risk assessment, decision-making, game theory, and data analysis. Understanding probability allows us to make informed choices in the face of uncertainty and to quantify the degree of confidence we can have in certain outcomes.
Probability MCQs with answers
1) What is the definition of probability?
a) The measure of how likely an event will occur.
b) The study of numbers and their properties.
c) A method of solving complex equations.
d) The process of counting occurrences.
2) In a fair six-sided die, what is the probability of rolling an even number?
a) 1/6
b) 1/3
c) 1/2
d) 2/3
3) A bag contains 5 red marbles and 3 blue marbles. What is the probability of drawing a red marble without replacement?
a) 5/8
b) 5/6
c) 5/7
d) 3/8
4) The probability of an event happening is 0.3. What is the probability of the event not happening?
a) 0.7
b) 0.3
c) 0.5
d) 1.0
5) If two dice are rolled, what is the probability of getting a sum of 7?
a) 1/12
b) 1/6
c) 1/9
d) 1/4
6) A box contains 8 red balls and 4 blue balls. What is the probability of randomly selecting a blue ball and then, on the next draw, selecting a red ball (without replacement)?
a) 1/12
b) 1/8
c) 1/6
d) 1/15
7) In a deck of cards, what is the probability of drawing a heart or a diamond?
a) 1/13
b) 1/2
c) 1/4
d) 1/26
8) If the probability of event A is 0.6 and the probability of event B is 0.4, what is the probability of both events A and B occurring?
a) 0.2
b) 0.24
c) 0.34
d) 0.14
9) A spinner is divided into 6 equal sections numbered 1 to 6. What is the probability of landing on an even number?
a) 2/3
b) 1/3
c) 1/2
d) 1/6
10) In a group of 30 students, 15 study Physics, 10 study Chemistry, and 5 study both. What is the probability that a randomly chosen student studies either Physics or Chemistry?
a) 1/3
b) 2/3
c) 1/2
d) 4/5
11) A jar contains 12 red balls and 8 green balls. If two balls are drawn randomly without replacement, what is the probability that both balls are green?
a) 2/29
b) 4/35
c) 1/4
d) 4/35
12) What is the probability of rolling a number greater than 4 on a standard six-sided die?
a) 1/6
b) 1/2
c) 1/3
d) 1/4
13) In a bag, there are 5 black balls and 3 white balls. What is the probability of drawing a black ball on the first draw and a white ball on the second draw (without replacement)?
a) 5/24
b) 3/8
c) 1/8
d) 5/16
14) If the probability of rain on any given day is 0.3, what is the probability of no rain on that day?
a) 0.7
b) 0.5
c) 0.3
d) 1.0
15) Two dice are rolled. What is the probability of getting a sum greater than 9?
a) 1/6
b) 1/4
c) 1/12
d) 1/3
16) A bag contains 6 red marbles and 4 blue marbles. If two marbles are drawn randomly with replacement, what is the probability of getting two red marbles?
a) 1/10
b) 3/20
c) 1/6
d) 3/10
17) If two cards are drawn successively from a standard deck of 52 cards, what is the probability of drawing a king on the first draw and a queen on the second draw (without replacement)?
a) 1/51
b) 1/26
c) 1/52
d) 1/104
18) In a town, it rains 60% of the days and is sunny 40% of the days. What is the probability that the next two days will be sunny?
a) 16/25
b) 4/5
c) 2/5
d) 16/100
19) A bag contains 8 balls, out of which 3 are red and 5 are blue. If 2 balls are drawn randomly without replacement, what is the probability of getting two blue balls?
a) 1/28
b) 1/4
c) 5/14
d) 5/16
20) A six-sided die is rolled twice. What is the probability of rolling two 6s?
a) 1/36
b) 1/18
c) 1/12
d) 1/6
Probability MCQs for competitive exams
1) If the probability of winning a game is 0.25, what is the probability of losing the game?
a) 0.5
b) 0.75
c) 0.25
d) 1.0
2) A bag contains 10 red balls and 6 blue balls. If three balls are drawn randomly with replacement, what is the probability of getting all red balls?
a) 1/8
b) 1/12
c) 1/2
d) 1/20
3) In a deck of cards, what is the probability of drawing a black card or a face card (jack, queen, or king)?
a) 1/4
b) 1/2
c) 1/3
d) 3/4
4) A box contains 5 red balls, 3 blue balls, and 2 green balls. If two balls are drawn randomly without replacement, what is the probability of getting a red ball and then a blue ball?
a) 5/20
b) 1/10
c) 3/10
d) 5/18
5) If the probability of an event is 0.9, what is the probability of its complement?
a) 0.1
b) 0.5
c) 1.0
d) 0.9
6) A bag contains 12 green balls and 8 red balls. If two balls are drawn randomly without replacement, what is the probability of getting two red balls?
a) 4/35
b) 2/25
c) 4/15
d) 8/35
7) In a group of 50 students, 30 like pizza, 20 like burgers, and 10 like both. What is the probability that a randomly chosen student likes either pizza or burgers?
a) 3/5
b) 2/5
c) 1/5
d) 1/2
8) If a card is drawn from a standard deck of 52 cards, what is the probability of drawing a heart?
a) 1/13
b) 1/52
c) 1/4
d) 1/2
9) Two dice are rolled. What is the probability of getting a sum of 5 or 6?
a) 1/6
b) 1/4
c) 1/3
d) 1/2
10) A bag contains 6 red balls and 4 blue balls. If two balls are drawn randomly without replacement, what is the probability of getting one red ball and one blue ball?
a) 2/5
b) 6/10
c) 3/10
d) 4/10
11) In a group of 40 people, 25 prefer tea and 20 prefer coffee. What is the probability that a randomly chosen person prefers either tea or coffee?
a) 9/40
b) 11/40
c) 1/2
d) 3/8
12) If the probability of drawing a red card from a standard deck of cards is 1/4, what is the probability of drawing a non-red card?
a) 1/4
b) 3/4
c) 1/2
d) 2/3
13) A bag contains 8 white balls and 6 black balls. If two balls are drawn randomly without replacement, what is the probability of getting two black balls?
a) 1/14
b) 3/14
c) 3/13
d) 2/13
14) If the probability of an event is 0.2, what is the probability of its complement?
a) 0.2
b) 0.8
c) 0.5
d) 1.0
15) A box contains 10 red balls and 5 blue balls. If two balls are drawn randomly without replacement, what is the probability of getting two blue balls?
a) 1/21
b) 1/20
c) 5/27
d) 1/15
16) In a group of 50 students, 30 play football, 20 play basketball, and 10 play both. What is the probability that a randomly chosen student plays either football or basketball?
a) 3/10
b) 1/5
c) 3/5
d) 1/2
17) If two dice are rolled, what is the probability of getting a sum of 10 or 11?
a) 1/6
b) 1/12
c) 1/9
d) 1/3
18) A bag contains 7 green balls and 5 yellow balls. If two balls are drawn randomly without replacement, what is the probability of getting one green ball and one yellow ball?
a) 5/12
b) 35/144
c) 1/12
d) 7/60
19) A box contains 10 identical pens, out of which 3 are defective. If 2 pens are drawn randomly with replacement, what is the probability of getting two defective pens?
a) 9/100
b) 9/50
c) 3/100
d) 3/50
20) In a bag, there are 12 red balls and 8 blue balls. What is the probability of drawing a blue ball on the first draw and a red ball on the second draw (without replacement)?
a) 1/10
b) 4/95
c) 1/8
d) 2/15
What is probability?
Probability is a mathematical concept used to measure the likelihood of an event occurring. It quantifies uncertainty and models random outcomes in various fields, including science, finance, and statistics.
How is probability calculated?
Probability is calculated by dividing the number of favorable outcomes by the total number of possible outcomes in a given sample space. It ranges from 0 to 1, where 0 indicates an impossible event, and 1 represents a certain event.
What are independent events?
Two events are independent if the occurrence of one event does not affect the likelihood of the other event happening. Mathematically, this means P(A ∩ B) = P(A) * P(B).
What is conditional probability?
Conditional probability measures the likelihood of an event occurring given that another event has already happened. It is denoted by P(A|B) and is calculated as P(A ∩ B) / P(B) (if P(B) > 0).
How is Bayes’ theorem applied?
Bayes’ theorem is used to update the probability of an event based on new evidence. It relates the conditional probabilities of two events in opposite orders: P(A|B) = P(B|A) * P(A) / P(B). It has applications in various fields, including medical diagnosis, machine learning, and data analysis.