Probability

35 solved questions from Class 12 Mathematics, each with the full working. Tap any question to see the step-by-step solution.

• 1 What is the domain, range, or fundamental theorem introduced in 'Probability' (Class 12 Mathematics)? medium MCQ Functions & Theorems
• 2 What is the derivative or integral formula associated with 'Probability'? medium MCQ Calculus & Formulae
• 3 How do you find the solution set or optimum value in 'Probability'? medium MCQ Problem Solving & Optimization
• 4 A standard deck has 52 cards. What is the probability of drawing a face card (Jack, Queen or King)? easy MCQ 1 mark Probability of a single event.
• 5 Two standard dice are rolled. What is the probability that the sum of the two numbers is at least 8? medium 2 marks Discrete random variables – dice sum probability.
• 6 A bag contains 4 red, 5 blue and 6 green balls. Three balls are drawn without replacement. What is the probability that exactly two of them are red and one is green? medium 3 marks Hypergeometric probability.
• 7 The continuous random variable Y has probability density function f(y)=k,y for 0le yle2 and f(y)=0 elsewhere. Find the value of k and the expected value E[Y]. hard 4 marks Normalizing a continuous density and finding expected value.
• 8 A bag contains 3 white, 5 black and 2 red balls. Two balls are drawn successively without replacement. Find the probability that both drawn balls are black given that the first drawn ball is black. medium 3 marks Conditional probability
• 9 A fair coin is tossed three times. Let X denote the number of heads obtained. (i) Write the probability distribution of X. (ii) Find the expected value E[X]. medium 2 marks Discrete random variable and expectation
• 10 A jar contains 2 red, 3 green and 5 blue marbles. One marble is drawn at random and not replaced. What is the probability that the second marble drawn is green given that the first marble drawn was blue? easy MCQ 1 mark Conditional probability
• 11 The lifetime T (in years) of a certain type of battery follows an exponential distribution with pdf f(t)=λ e^(-λ t) for t≥ 0, where λ =0.2 per year. (i) Find the probability that a randomly selected battery lasts more than 5 years. (ii) Find the mean lifetime E[T]. medium 3 marks Exponential distribution and its expectation
• 12 In a town, 1% of the population suffers from disease D. A diagnostic test for D has the following properties: it correctly identifies a diseased person (positive result) 95% of the time, and correctly identifies a healthy person (negative result) 90% of the time. (i) If a randomly selected person tests positive once, what is the probability that the person actually has the disease? (ii) If the same person undergoes two independent tests and both return positive, what is the updated probability of having the disease? hard 4 marks Bayes theorem with prior probabilities
• 13 Two cards are drawn successively without replacement from a well‑shuffled standard deck of 52 cards. Given that the first card drawn is a heart, find the probability that the second card drawn is a king. medium 2 marks Conditional probability without replacement
• 14 A standard deck of 52 playing cards is shuffled. One card is drawn at random. What is the probability that the card is an ace? easy MCQ 1 mark Simple probability
• 15 An exam consists of 10 multiple‑choice questions. Each question has four options of which exactly one is correct. A student guesses each answer independently. Let X be the number of correct answers. hard 3 marks Binomial distribution (mean & variance)
• 16 Let Y be a continuous random variable with probability density function f(y)=k y^2 for 0≤ y≤ 1 and f(y)=0 elsewhere. hard 4 marks Finding constant, probability, expectation and variance for a continuous pdf
• 17 Three fair dice are rolled simultaneously. Find the probability that the sum of the numbers shown on the dice is exactly 10. medium 2 marks Discrete uniform distribution – dice sums
• 18 A bag contains 8 black, 5 white, and 7 green balls. Two balls are drawn without replacement. Calculate the probability that exactly one of the two drawn balls is green. medium 2 marks Hypergeometric distribution – single draw without replacement
• 19 Two cards are drawn successively without replacement from a standard deck of 52 cards. What is the probability that both cards are of the same suit and at least one of them is an ace? hard 4 marks Counting combinations – probability with multiple conditions
• 20 Let X be a continuous random variable with probability density function f(x)=k,x^2 for 0le xle 2 and f(x)=0 otherwise. hard 5 marks Continuous probability distribution and moments
• 21 Two cards are drawn successively without replacement from a standard deck of 52 cards. What is the probability that the first card is a spade and the second card is a heart? medium 2 marks Conditional probability without replacement
• 22 An urn contains 6 red, 4 blue, and 5 green balls. Two balls are drawn successively without replacement. What is the probability that both balls drawn are red? easy MCQ 1 mark Probability of dependent events
• 23 Let X be a continuous random variable with pdf f(x)=3x^2 for 0le xle1 and f(x)=0 otherwise. Find the expected value E[X] and the variance Var(X). medium 2 marks Mean and variance of continuous distributions
• 24 Two fair dice are rolled simultaneously. What is the probability that the sum of the numbers shown on the dice is divisible by 4 and at least one of the dice shows a 6? hard 3 marks Combinatorial probability with multiple constraints
• 25 The lifetime T (in years) of a certain type of battery follows an exponential distribution with pdf f(t)=λ e^(-λ t) for t≥ 0, where λ =0.2 per year. What is the probability that a randomly selected battery lasts more than its mean lifetime? hard MCQ 4 marks Exponential distribution mean and tail probability
• 26 In a certain town, 2% of the population suffers from disease D. A diagnostic test for D has sensitivity 90% (it reports positive for a diseased person) and specificity 85% (it reports negative for a healthy person).\n(i) If a randomly selected person tests positive once, what is the probability that he actually has the disease?\n(ii) If the same person undergoes two independent tests and both return positive, what is the updated probability that he has the disease? hard 4 marks Bayes theorem with multiple tests
• 27 A box contains 5 red, 7 blue, and 8 yellow marbles. Two marbles are drawn successively without replacement. Find the probability that the second marble is blue given that the first marble drawn is red. medium 2 marks Conditional probability without replacement
• 28 A fair coin is tossed three times. What is the probability of getting exactly two heads? easy 1 mark Binomial probability for small n
• 29 In a town, 0.3% of the population has disease D. A diagnostic test has sensitivity 80% and specificity 95%. If a person undergoes two independent tests and both return positive, what is the probability that the person actually has disease D? hard 5 marks Bayes theorem with multiple tests
• 30 Two cards are drawn successively without replacement from a standard 52‑card deck. What is the probability that the second card is a king given that the first card drawn is a heart? easy MCQ 3 marks Conditional probability with dependent draws
• 31 Let X be a continuous random variable with probability density function f(x)=2x for 0le xle 1 and f(x)=0 otherwise. Find the expected value E[X] and variance Var(X). medium 4 marks Expectation and variance of continuous distribution
• 32 A fair die is rolled five times. What is the probability of obtaining exactly three sixes? medium 4 marks Binomial distribution probability
• 33 Two cards are drawn successively without replacement from a standard 52‑card deck. What is the probability that the second card is a spade given that the first card drawn is a heart? medium 2 marks Conditional probability without replacement
• 34 A fair coin is tossed 4 times. What is the probability of obtaining exactly 2 heads? easy MCQ 1 mark Binomial probability calculation
• 35 Let X be a continuous random variable with probability density function f(x)=4x^3 for 0≤x≤1 and f(x)=0 otherwise. Find the expected value E[X] and variance Var(X). hard 5 marks Expectation and variance of continuous variable