1 P (A) = = , 100 10 2 1 P (A ∩ B) = = 100 50 PROBABILITY 263 P (A ∩ B) 1 10 1 P (B | A) = = × = P (A) 50 1 5 Thus the probability of the picked up bulb of its being defective, if it is red, is .
For a loaded die, the probabilities of outcomes are given as under: P(1) = P(2) = 0.2, P(3) = P(5) = P(6) = 0.1 and P(4) = 0.3. The die is thrown two times. Let A and B be the events, ‘same number each time’, and ‘a total score is 10 or more’, respectively. Determine whether or not A and B are independent.
Refer to Exercise 1 above. If the die were fair, determine whether or not the events A and B are independent.
The probability that at least one of the two events A and B occurs is 0.6. If A and B occur simultaneously with probability 0.3, evaluate P( A ) + P( B ).
A bag contains 5 red marbles and 3 black marbles. Three marbles are drawn one by one without replacement. What is the probability that at least one of the three marbles drawn be black, if the first marble is red? 272 MATHEMATICS
Two dice are thrown together and the total score is noted. The events E, F and G are ‘a total of 4’, ‘a total of 9 or more’, and ‘a total divisible by 5’, respectively. Calculate P(E), P(F) and P(G) and decide which pairs of events, if any, are independent.
Explain why the experiment of tossing a coin three times is said to have binomial distribution. 1 1 1
A and B are two events such that P(A) = , P(B) = and P(A ∩ B)= . 2 3 4 Find :
- (i)P(A|B)
- (ii)P(B|A)
- (iii)P(A'|B)
- (iv)P(A'|B') 2 1 1
Three events A, B and C have probabilities , and , respectively. Given 5 3 2 1 1 that P(A ∩ C) = and P(B ∩ C) = , find the values of P(C | B) and P(A' ∩ C'). 5 4
Let E1 and E2 be two independent events such that p(E1) = p1 and P(E2) = p2. Describe in words of the events whose probabilities are:
- (i)p1 p2
- (ii)(1–p1) p2
- (iii)1–(1–p1)(1–p2)
- (iv)p1 + p2 – 2p1p2
A discrete random variable X has the probability distribution given as below: X 0.5 1 1.5 2 P(X) k k2 2k 2 k
- (i)Find the value of k
- (ii)Determine the mean of the distribution.
Prove that
- (i)P(A) = P(A ∩ B) + P(A ∩ B )
- (ii)P(A ∪ B) = P(A ∩ B) + P(A ∩ B ) + P( A ∩ B)
If X is the number of tails in three tosses of a coin, determine the standard deviation of X.
In a dice game, a player pays a stake of Re1 for each throw of a die. She receives Rs 5 if the die shows a 3, Rs 2 if the die shows a 1 or 6, and nothing PROBABILITY 273 otherwise. What is the player’s expected profit per throw over a long series of throws?
Three dice are thrown at the sametime. Find the probability of getting three two’s, if it is known that the sum of the numbers on the dice was six.
Suppose 10,000 tickets are sold in a lottery each for Re 1. First prize is of Rs 3000 and the second prize is of Rs. 2000. There are three third prizes of Rs. 500 each. If you buy one ticket, what is your expectation.
A bag contains 4 white and 5 black balls. Another bag contains 9 white and 7 black balls. A ball is transferred from the first bag to the second and then a ball is drawn at random from the second bag. Find the probability that the ball drawn is white.
Bag I contains 3 black and 2 white balls, Bag II contains 2 black and 4 white balls. A bag and a ball is selected at random. Determine the probability of selecting a black ball.
A box has 5 blue and 4 red balls. One ball is drawn at random and not replaced. Its colour is also not noted. Then another ball is drawn at random. What is the probability of second ball being blue?
Four cards are successively drawn without replacement from a deck of 52 playing cards. What is the probability that all the four cards are kings?
A die is thrown 5 times. Find the probability that an odd number will come up exactly three times.
Ten coins are tossed. What is the probability of getting at least 8 heads?
The probability of a man hitting a target is 0.25. He shoots 7 times. What is the probability of his hitting at least twice?
A lot of 100 watches is known to have 10 defective watches. If 8 watches are selected (one by one with replacement) at random, what is the probability that there will be at least one defective watch? 274 MATHEMATICS
Consider the probability distribution of a random variable X: X 0 1 2 3 4 P(X) 0.1 0.25 0.3 0.2 0.15 X Calculate
- (i)V
- (ii)Variance of X. 2
The probability distribution of a random variable X is given below: X 0 1 2 3 k k k P(X) k 2 4 8
- (i)Determine the value of k.
- (ii)Determine P(X ≤ 2) and P(X > 2)
- (iii)Find P(X ≤ 2) + P (X > 2).
For the following probability distribution determine standard deviation of the random variable X. X 2 3 4 P(X) 0.2 0.5 0.3
A biased die is such that P(4) = and other scores being equally likely. The die is tossed twice. If X is the ‘number of fours seen’, find the variance of the random variable X.
A die is thrown three times. Let X be ‘the number of twos seen’. Find the expectation of X.
Two biased dice are thrown together. For the first die P(6) = , the other scores being equally likely while for the second die, P(1) = and the other scores are PROBABILITY 275 equally likely. Find the probability distribution of ‘the number of ones seen’.
Two probability distributions of the discrete random variable X and Y are given below. X 0 1 2 3 Y 0 1 2 3 1 2 1 1 1 3 2 1 P(X) P(Y) 5 5 5 5 5 10 5 10 Prove that E(Y2) = 2 E(X).
A factory produces bulbs. The probability that any one bulb is defective is and they are packed in boxes of 10. From a single box, find the probability that
- (i)none of the bulbs is defective
- (ii)exactly two bulbs are defective
- (iii)more than 8 bulbs work properly
Suppose you have two coins which appear identical in your pocket. You know that one is fair and one is 2-headed. If you take one out, toss it and get a head, what is the probability that it was a fair coin?
Suppose that 6% of the people with blood group O are left handed and 10% of those with other blood groups are left handed 30% of the people have blood group O. If a left handed person is selected at random, what is the probability that he/she will have blood group O?
Two natural numbers r, s are drawn one at a time, without replacement from the set S= {1, 2, 3, ...., n } . Find P [ r ≤ p|s ≤ p ] , where p ∈ S.
Find the probability distribution of the maximum of the two scores obtained when a die is thrown twice. Determine also the mean of the distribution.
The random variable X can take only the values 0, 1, 2. Given that P(X = 0) = P (X = 1) = p and that E(X2) = E[X], find the value of p. 276 MATHEMATICS
Find the variance of the distribution: x 0 1 2 3 4 5 1 5 2 1 1 1 P(x) 6 18 9 6 9 18
A and B throw a pair of dice alternately. A wins the game if he gets a total of 6 and B wins if she gets a total of 7. It A starts the game, find the probability of winning the game by A in third throw of the pair of dice.
Two dice are tossed. Find whether the following two events A and B are independent: A = { (x, y ) : x +y =11} B = { (x, y ) : x ≠ 5} where (x, y) denotes a typical sample point.
An urn contains m white and n black balls. A ball is drawn at random and is put back into the urn along with k additional balls of the same colour as that of the ball drawn. A ball is again drawn at random. Show that the probability of drawing a white ball now does not depend on k.
Three bags contain a number of red and white balls as follows: Bag 1 : 3 red balls, Bag 2 : 2 red balls and 1 white ball Bag 3 : 3 white balls. The probability that bag i will be chosen and a ball is selected from it is , i = 1, 2, 3. What is the probability that
- (i)a red ball will be selected?
- (ii)a white ball is selected?
Refer to Question 41 above. If a white ball is selected, what is the probability that it came from
A shopkeeper sells three types of flower seeds A1, A2 and A3. They are sold as a mixture where the proportions are 4:4:2 respectively. The germination rates of the three types of seeds are 45%, 60% and 35%. Calculate the probability
- (i)of a randomly chosen seed to germinate PROBABILITY 277
- (ii)that it will not germinate given that the seed is of type A3,
- (iii)that it is of the type A2 given that a randomly chosen seed does not germinate.
A letter is known to have come either from TATA NAGAR or from CALCUTTA. On the envelope, just two consecutive letter TA are visible. What is the probability that the letter came from TATA NAGAR.
There are two bags, one of which contains 3 black and 4 white balls while the other contains 4 black and 3 white balls. A die is thrown. If it shows up 1 or 3, a ball is taken from the Ist bag; but it shows up any other number, a ball is chosen from the second bag. Find the probability of choosing a black ball.
There are three urns containing 2 white and 3 black balls, 3 white and 2 black balls, and 4 white and 1 black balls, respectively. There is an equal probability of each urn being chosen. A ball is drawn at random from the chosen urn and it is found to be white. Find the probability that the ball drawn was from the second urn.
By examining the chest X ray, the probability that TB is detected when a person is actually suffering is 0.99. The probability of an healthy person diagnosed to have TB is 0.001. In a certain city, 1 in 1000 people suffers from TB. A person is selected at random and is diagnosed to have TB. What is the probability that he actually has TB?
An item is manufactured by three machines A, B and C. Out of the total number of items manufactured during a specified period, 50% are manufactured on A, 30% on B and 20% on C. 2% of the items produced on A and 2% of items produced on B are defective, and 3% of these produced on C are defective. All the items are stored at one godown. One item is drawn at random and is found to be defective. What is the probability that it was manufactured on machine A?
Let X be a discrete random variable whose probability distribution is defined as follows: k ( x + 1) for x = 1, 2,3, 4 P (X = x) = 2kx for x = 5,6,7 0 otherwise 278 MATHEMATICS where k is a constant. Calculate
- (i)the value of k
- (ii)E (X)
- (iii)Standard deviation of X.
The probability distribution of a discrete random variable X is given as under: X 1 2 4 2A 3A 5A 1 1 3 1 1 1 P(X) 2 5 25 10 25 25 Calculate :
- (i)The value of A if E(X) = 2.94
- (ii)Variance of X.
The probability distribution of a random variable x is given as under: kx 2 for x = 1, 2,3 2kx for x = 4,5,6 P( X = x ) = 0 otherwise where k is a constant. Calculate
- (i)E(X)
- (ii)E (3X2)
- (iii)P(X ≥ 4)
A bag contains (2n + 1) coins. It is known that n of these coins have a head on both sides where as the rest of the coins are fair. A coin is picked up at random from the bag and is tossed. If the probability that the toss results in a head is , determine the value of n.
Two cards are drawn successively without replacement from a well shuffled deck of cards. Find the mean and standard variation of the random variable X where X is the number of aces.
A die is tossed twice. A ‘success’ is getting an even number on a toss. Find the variance of the number of successes.
There are 5 cards numbered 1 to 5, one number on one card. Two cards are drawn at random without replacement. Let X denote the sum of the numbers on two cards drawn. Find the mean and variance of X. PROBABILITY 279
to 82. 4 7 56. If P(A) = , and P(A ∩ B) = , then P(B | A) is equal to 5 10 1 1 7 17
- (A)
- (B)
- (C)
- (D)10 8 8 20 7 17
If P(A ∩ B) = and P(B) = , then P (A | B) equals 10 20 14 17 7 1
- (A)
- (B)
- (C)
- (D)17 20 8 8 3 2 3
If P(A) = , P (B) = and P(A∪B) = , then P (B | A) + P (A | B) equals 10 5 5 1 1 5 7 (A) (B) (C) (D) 4 3 12 2 2 3 1
If P(A) = , P(B) = and P (A ∩ B) = , then P(A ′ | B′ ).P(B ' | A ') is equal 5 10 5 5 5 25
- (A)
- (B)
- (C)
- (D)1 6 7 42 1 1 1
If A and B are two events such that P(A) = , P(B) = , P(A/B)= , then 2 3 4 P(A ′ ∩ B′ ) equals 1 3 1 3
- (A)
- (B)
- (C)
- (D)12 4 4 16 280 MATHEMATICS
If P(A) = 0.4, P(B) = 0.8 and P(B | A) = 0.6, then P(A ∪ B) is equal to
- (A)0.24
- (B)0.3
- (C)0.48
- (D)0.96
If A and B are two events and A ≠ φ, B ≠ φ, then P(A ∩ B)
- (A)P(A | B) = P(A).P(B)
- (B)P(A | B) = P(B)
- (C)P(A | B).P(B | A)=1
- (D)P(A | B) = P(A) | P(B)
A and B are events such that P(A) = 0.4, P(B) = 0.3 and P(A ∪ B) = 0.5. Then P (B′ ∩ A) equals 2 1 3 1
You are given that A and B are two events such that P(B)= , P(A | B) = and 5 2 P(A ∪ B) = , then P(A) equals 3 1 1 3
In Exercise 64 above, P(B | A′ ) is equal to 1 3 1 3
- (A)
- (B)
- (C)
- (D)5 10 2 5 3 1 4
If P(B) = , P(A | B) = and P(A ∪ B) = , then P(A ∪ B )′ + P( A′ ∪ B) = 5 2 5 1 4 1
- (A)
- (B)
- (C)
- (D)1 5 5 2 PROBABILITY 281 7 9 4
Let P(A) = , P(B) = and P(A ∩ B) = . Then P( A′ | B) is equal to 13 13 13 6 4 4 5
If A and B are such events that P(A) > 0 and P(B) ≠ 1, then P( A′ | B′ ) equals.
- (A)1 – P(A | B)
- (B)1– P( A′ | B) 1–P(A ∪ B)
- (C)P(B')
- (D)P( A′ ) | P( B′ ) 3 4
If A and B are two independent events with P(A) = and P(B) = , then 5 9 P( A′ ∩ B′ ) equals 4 8 1 2
If two events are independent, then
- (A)they must be mutually exclusive
- (B)the sum of their probabilities must be equal to 1
- (C)(A) and (B) both are correct
- (D)None of the above is correct 3 5 3
Let A and B be two events such that P(A) = , P(B) = and P(A ∪ B) = . 8 8 4 Then P(A | B).P( A′ | B) is equal to 2 3 3 6
If the events A and B are independent, then P(A ∩ B) is equal to 282 MATHEMATICS
- (A)P (A) + P
- (B)(B) P(A) – P(B)
- (C)P (A) . P(B)
- (D)P(A) | P(B)
Two events E and F are independent. If P(E) = 0.3, P(E ∪ F) = 0.5, then P(E | F)–P(F | E) equals 2 3 1 1
A bag contains 5 red and 3 blue balls. If 3 balls are drawn at random without replacement the probability of getting exactly one red ball is 45 135 15 15
- (A)
- (B)
- (C)
- (D)196 392 56 29
Refer to Question 74 above. The probability that exactly two of the three balls were red, the first ball being red, is 1 4 15 5
Three persons, A, B and C, fire at a target in turn, starting with A. Their probability of hitting the target are 0.4, 0.3 and 0.2 respectively. The probability of two hits
- (A)0.024
- (B)0.188
- (C)0.336
- (D)0.452
Assume that in a family, each child is equally likely to be a boy or a girl. A family with three children is chosen at random. The probability that the eldest child is a girl given that the family has at least one girl is 1 1 2 4
A die is thrown and a card is selected at random from a deck of 52 playing cards. The probability of getting an even number on the die and a spade card is 1 1 1 3
- (A)
- (B)
- (C)
- (D)2 4 8 4 PROBABILITY 283
A box contains 3 orange balls, 3 green balls and 2 blue balls. Three balls are drawn at random from the box without replacement. The probability of drawing 2 green balls and one blue ball is 3 2 1 167
A flashlight has 8 batteries out of which 3 are dead. If two batteries are selected without replacement and tested, the probability that both are dead is 33 9 1 3
Eight coins are tossed together. The probability of getting exactly 3 heads is 1 7 5 3
Two dice are thrown. If it is known that the sum of numbers on the dice was less than 6, the probability of getting a sum 3, is 1 5 1 2
Which one is not a requirement of a binomial distribution?
- (A)There are 2 outcomes for each trial
- (B)There is a fixed number of trials
- (C)The outcomes must be dependent on each other
- (D)The probability of success must be the same for all the trials
Two cards are drawn from a well shuffled deck of 52 playing cards with replacement. The probability, that both cards are queens, is 1 1 1 1 1 1 1 4
- (A)×
- (B)+
- (C)×
- (D)× 13 13 13 1 3 13 17 13 51
The probability of guessing correctly at least 8 out of 10 answers on a true-false type examination is 284 MATHEMATICS 7 7 45 7
- (A)
- (B)
- (C)
- (D)64 128 1024 41
The probability that a person is not a swimmer is 0.3. The probability that out of 5 persons 4 are swimmers is
- (A)5C4 (0.7)4 (0.3)
- (B)5C1 (0.7) (0.3)4
- (C)5C4 (0.7) (0.3)4
- (D)(0.7)4 (0.3)
The probability distribution of a discrete random variable X is given below: X 2 3 4 5 5 7 9 11 P(X) k k k k The value of k is
For the following probability distribution: X –4 –3 –2 –1 0 P(X) 0.1 0.2 0.3 0.2 0.2 E(X) is equal to :
For the following probability distribution X 1 2 3 4 1 1 3 2 P (X) 10 5 10 5 E(X2) is equal to
Suppose a random variable X follows the binomial distribution with parameters n and p, where 0 < p < 1. If P(x = r) / P(x = n–r) is independent of n and r, then p equals PROBABILITY 285 1 1 1 1
In a college, 30% students fail in physics, 25% fail in mathematics and 10% fail in both. One student is chosen at random. The probability that she fails in physics if she has failed in mathematics is 1 2 9 1
A and B are two students. Their chances of solving a problem correctly are 1 1 and , respectively. If the probability of their making a common error is, 4 20 and they obtain the same answer, then the probability of their answer to be correct is 1 1 13 10
A box has 100 pens of which 10 are defective. What is the probability that out of a sample of 5 pens drawn one by one with replacement at most one is defective? 5 4 5 5 4 9 1 9 1 9 9 1 9
- (A)
- (B)
- (C)
- (D) + 10 2 10 2 10 10 2 10 State True or False for the statements in each of the Exercises 94 to 103.
Let P(A) > 0 and P(B) > 0. Then A and B can be both mutually exclusive and independent.
If A and B are independent events, then A′ and B′ are also independent.
If A and B are mutually exclusive events, then they will be independent also.
Two independent events are always mutually exclusive.
If A and B are two independent events then P(A and B) = P(A).P(B). 286 MATHEMATICS
Another name for the mean of a probability distribution is expected value.
If A and B′ are independent events, then P(A' ∪ B) = 1 – P (A) P(B')
If A and B are independent, then P (exactly one of A, B occurs) = P (A) P (B′ ) + P ( B) P ( A′ )
If A and B are two events such that P(A) > 0 and P(A) + P(B) >1, then P (B′) P(B | A) ≥ 1− P(A)
If A, B and C are three independent events such that P(A) = P(B) = P(C) = p, then P (At least two of A, B, C occur) = 3 p 2 − 2 p 3 Fill in the blanks in each of the following questions:
If A and B are two events such that P (A | B) = p, P(A) = p, P(B) = and P(A ∪ B)= , then p = _____
If A and B are such that 2 5 P(A' ∪ B') = and P(A ∪ B)= , 3 9 then P(A') + P(B') = ..................
If X follows binomial distribution with parameters n = 5, p and P (X = 2) = 9, P (X = 3), then p = ___________
Let X be a random variable taking values x1, x2,..., xn with probabilities p1, p2, ..., pn, respectively. Then var (X) = ________
Let A and B be two events. If P(A | B) = P(A), then A is ___________ of B.