CBSE Class 12 Mathematics Probability VBQs Set C

Short Answer Questions:

Question. The probability of simultaneous occurrence of atleast one of two events A and B is \( p \). If the probability that exactly one of A, B occurs is \( q \), then prove that \( P(A) + P(B) = 2p - q \).
Answer: Given \( P(A \cup B) = p \) and \( P(A \cap \bar{B}) + P(\bar{A} \cap B) = q \).
\( \because P(A \cup B) = P(A \cap \bar{B}) + P(\bar{A} \cap B) + P(A \cap B) \)
\( \Rightarrow p = q + P(A \cap B) \Rightarrow P(A \cap B) = p - q \).
Now, \( P(A) + P(B) = P(A \cup B) + P(A \cap B) = p + (p - q) = 2p - q \). Hence proved.

Question. A and B throw a pair of dice alternately. A wins the game if he gets a total of 7 and B wins the game if he gets a total of 10. If A starts the game, then find the probability that B wins.
Answer: \( \frac{5}{17} \)

Question. A random variable X has the following probability distribution:

Find the value of C and also calculate mean of the distribution. 
Answer: \( C = \frac{1}{10} \), Mean \( = 2.66 \)

Question. A and B throw a pair of dice alternately, till one of them gets a total of 10 and wins the game. Find their respective probabilities of winning, if A starts first. 
Answer: \( \frac{12}{23}, \frac{11}{23} \)

Question. A box has 20 pens of which 2 are defective. Calculate the probability that out of 5 pens drawn one by one with replacement, at most 2 are defective. 
Answer: \( \frac{99144}{100000} \)

Question. Let, X denote the number of colleges where you will apply after your results and \( P(X = x) \) denotes your probability of getting admission in \( x \) number of colleges. It is given that
\( P(X = x) = \begin{cases} kx, & \text{if } x = 0 \text{ or } 1 \\ 2kx, & \text{if } x = 2 \\ k(5-x), & \text{if } x = 3 \text{ or } 4 \\ 0, & \text{if } x > 4 \end{cases} \)
where \( k \) is a positive constant. Find the value of \( k \). Also, find the probability that you will get admission in (i) exactly one college (ii) at most 2 colleges (iii) at least 2 colleges.
Answer: \( k = \frac{1}{8} \); (i) \( \frac{1}{8} \), (ii) \( \frac{5}{8} \), (iii) \( \frac{7}{8} \)

Question. There are 4 cards numbered 1, 3, 5 and 7, one number on one card. Two cards are drawn at random without replacement. Let X denote the sum of the numbers on the two drawn cards. Find the mean and variance of X. 
Answer: Mean \( = 8 \); Variance \( = \frac{20}{3} \)

Question. P speaks truth in 70% of the cases and Q in 80% of the cases. In what percent of cases are they likely to agree in stating the same fact? 
Answer: 62%

Question. In a group of 50 scouts in a camp, 30 are well trained in first aid techniques while the remaining are well trained in hospitality but not in first aid. Two scouts are selected at random from the group. Find the probability distribution of number of selected scouts who are well trained in first aid. Find the mean of the distribution also. 
Answer: Mean \( = \frac{294}{245} \). Probability distribution:

Question. Of the students in a school; it is known that 30% have 100% attendance and 70% students are irregular. Previous year results report that 70% of all students who have 100% attendance attain A grade and 10% irregular students attain A grade in their annual examination. At the end of the year, one student is chosen at random from the school and he has A grade. What is the probability that the student has 100% attendance? 
Answer: \( \frac{3}{4} \)

Question. How many times should a man toss a fair coin so that the probability of having at least one head is more than 90%?
Answer: \( n \geq 4 \)

Question. Three numbers are selected at random (without replacement) from first six positive integers. Let X denote the largest of the three numbers obtained. Find the probability distribution of X. Also, find the mean and variance of the distribution. 
Answer: \( X \) takes values 3, 4, 5, 6. Mean \( = 5.25 \), Var \( = 0.6875 \).

Question. Suppose 10000 tickets are sold in a lottery each for ₹1. First prize is of ₹3000 and the second prize is of ₹2000. There are three third prizes of ₹500 each. If you buy one ticket, then what is your expectation? 
Answer: ₹ 0.65

Question. The probability that A hits a target is \( \frac{1}{3} \) and the probability that B hits it is \( \frac{2}{5} \). If each one of A and B shoots at the target, what is the probability that (i) the target is hit? (ii) exactly one of them hits the target? 
Answer: (i) \( \frac{3}{5} \), (ii) \( \frac{7}{15} \)

Question. A bag X contains 4 white balls and 2 black balls, while another bag Y contains 3 white balls and 3 black balls. Two balls are drawn (without replacement) at random from one of the bags and were found to be one white and one black. Find the probability that the balls were drawn from bag Y. 
Answer: \( \frac{9}{17} \)

Question. The probabilities of two students A and B coming to the school in time are \( \frac{3}{7} \) and \( \frac{5}{7} \) respectively. Assuming that the events, ‘A coming in time’ and ‘B coming in time’ are independent, find the probability of only one of them coming to the school in time. 
Answer: \( \frac{26}{49} \)

Long Answer Questions:

Question. Find the probability distribution of the random variable X, which denotes the number of doublets in four throws of a pair of dice. Hence, find the mean of the number of doublets (X). 
Answer: Mean \( = \frac{2}{3} \). Dist: \( P(X=r) = {}^4C_r (1/6)^r (5/6)^{4-r} \).

Question. An insurance company insured 3,000 scooters, 4,000 cars and 5,000 trucks. The probabilities of the accident involving a scooter, a car and a truck are 0.02, 0.03 and 0.04 respectively. One of the insured vehicles meet with an accident. Find the probability that it is a (a) scooter (b) car (c) truck. 
Answer: (a) \( \frac{3}{19} \), (b) \( \frac{6}{19} \), (c) \( \frac{10}{19} \)

Question. In a bolt factory, three machines A, B and C manufacture 25, 35 and 40 per cent of the total bolts manufactured. Of their output, 5, 4 and 2 per cent are defective respectively. A bolt is drawn at random and is found to be defective. Find the probability that it was manufactured by either machine A or C.
Answer: \( \frac{41}{69} \)

Question. In a factory which manufactures bolts, machines A, B and C manufacture respectively 30%, 50% and 20% of the bolts. Of their outputs 3, 4, 1 per cent respectively are defective bolts. A bolts is drawn at random from the product and is found to be defective. Find the probability that this is not manufactured by machine B. 
Answer: \( \frac{11}{31} \)

Question. Find the mean, the variance and the standard deviation of the number of doublets in three throws of a pair of dice. 
Answer: Mean \( = 1/2 \), Var \( = 5/12 \), SD \( = \sqrt{5/12} \).

Question. A, B and C throw a pair of dice in that order alternately till one of them gets a total of 9 and wins the game. Find their respective probabilities of winning, if A starts first.
Answer: \( \frac{81}{217}, \frac{72}{217}, \frac{64}{217} \)

Question. Bag A contains 3 red and 5 black balls, while bag B contains 4 red and 4 black balls. Two balls are transferred at random from bag A to bag B and then a ball is drawn from bag B at random. If the ball drawn from bag B is found to be red, find the probability that two red balls were transferred from A to B. 
Answer: \( \frac{3}{20} \)

Question. An urn contains 3 white and 6 red balls. Four balls are drawn one by one with replacement from the urn. Find the probability distribution of the number of red balls drawn. Also find mean and variance of the distribution.
Answer: Mean \( = 8/3 \), Var \( = 8/9 \).

Question. Five bad oranges are accidently mixed with 20 good ones. If four oranges are drawn one by one successively with replacement, then find the probability distribution of number of bad oranges drawn. Hence find the mean and variance of the distribution. 
Answer: Mean \( = 0.64 \), Var \( = 0.512 \).

Question. If A and B are two independent events such that \( P(\bar{A} \cap B) = \frac{2}{15} \) and \( P(A \cap \bar{B}) = \frac{1}{6} \), then find \( P(A) \) and \( P(B) \). 
Answer: \( P(A) = 1/5, P(B) = 1/6 \) or \( P(A) = 5/6, P(B) = 4/5 \).

Question. An urn contains 5 red and 2 black balls. Two balls are randomly drawn, without replacement. Let X represent the number of black balls drawn. What are the possible values of X? Is X a random variable? If yes, find the mean and variance of X.
Answer: Yes, values: 0, 1, 2. Mean \( = 4/7 \), Var \( = 40/147 \).

Question. A man is known to speak truth 3 out of 5 times. He throws a die and reports that it is 4. Find the probability that it is actually a 4. 
Answer: \( \frac{3}{13} \)

Question. There are two boxes I and II. Box I contains 3 red and 6 black balls. Box II contains 6 red and ‘n’ black balls. One of the two boxes, box I and box II is selected at random and a ball is drawn at random. The ball drawn is found to be red. If the probability that this red ball comes out from box II is \( \frac{3}{5} \), find the value of ‘n’. 
Answer: \( n = 4 \)

Question. A girl throws a die. If she gets a 5 or 6, she tosses a coin three times and notes the number of heads. If she gets 1, 2, 3 or 4, she tosses a coin two times and notes the number of heads obtained. If she obtained exactly two heads, what is the probability that she threw 1, 2, 3 or 4 with the die? 
Answer: \( \frac{4}{7} \)

Question. A bag contains 4 balls. Two balls are drawn at random and are found to be white. What is the probability that all balls are white? 
Answer: \( \frac{3}{5} \)

Question. Two groups are competing for the position on the Board of Directors of a corporation. The probabilities that the first and the second group will win are 0.6 and 0.4 respectively. Further, if the first group wins, the probability of introducing a new product is 0.7 and the corresponding probability is 0.3, if the second group wins. Find the probability that the new product was introduced by the second group. 
Answer: \( \frac{2}{9} \)