1. From 21 ticket marked 20 to 40 numerals, one is drawn at random, Find the chance that
it is multiple of 5.
2. If you twice flip a balanced coin, what is the probability of getting at least one head?
3.
If two dice are rolled, find the probability that the sum of the uppermost faces will (a)
equal 6 and ii) equal 8.
4. A number is drawn from the first 20 natural numbers. What is the probability that it is
a multiple of 3 or 5
5. What is the probability that a leap year, selected at random, will contain 53 Sundays?
6. What is the probability that a non-leap year, selected at random, will contain 53
Sundays?
7. Construct tree diagram for the following scenario. “Four aces are removed from a deck
of cards. A fair coin is tossed and one of aces is chosen”. Also determine the probability
of getting heads on the coin and the ace of the hearts?
8. Two dice are thrown together. Construct a tree diagram and then determine the
probability that both numbers are less than five.
9. A person wakes up late on average 3 days In every 5. If the person wakes up late, the
probability he is late to probability class is 9/10. If he does not wake up late, the
probability that he is late for probability class is 3/10. Determine on what percentage
of days does the person get to school on time? Generate Tree diagram and then
determine the probability
10. A box contains 5 white, 4 red and 6 green balls. Three balls are drawn at random. Find
the probability that a white, a read, and a green ball are drawn. ( Write down the sample
space also)
11. Four persons are choosen at random from a group containing 3 men, 2 women and 4
children. Show that the chances that exactly two of them will be children is 10/21.
12. Each coefficient in the equation ax2 + bx + c =0 is determined by throwing an ordinary
die. Find the probability that the equation will have real roots.
13. A person is asked to choose a four-digit ATM pin number. Each digit can be chosen
from 0 to 9. How many different possible pin numbers can the person choose.
14. If A and B are any two events. Show that P(A U B) = P(A) + P(B) – P( A Ω B) using
venn diagram.
15. Following Q.14. If A, B and C are any three events, then show that
P(AUBUC) = P(A) + P(B) + P(C) - [ P( A Ω B) + P ( B Ω C) + P( A Ω C) ] +
P [ A Ω B Ω C)
16. After conducting some experiment P(A) = 1/2 , P(B) = 1/3 and P (A Ω B) = 1/4.
Compute the following and also mention the type of law applied.
i)
P(𝐴Ω 𝐵)
ii)
P (𝐴 Ω B)
iii)
17. Three students A, B , C are in a swimming race. A and B have the same probability of
winning and each is twice as likely to win as C. Find the probability of winning of each
of the students.
18. Given that for the events A and B P( A+B) = 3/4 , P(AB) = 1/4 and P (𝐴) = 2/3. Find
P(B).
19. What is the difference between trail and event?
20. Explain and provide an example for the following
a) Equally Likely Events
b) Favourable Events
c) Mutually Exclusive/disjoint events
d) Exhaustive events
e) Independent events
f) Dependent events
g) Simple and compound events
21. A bag contains 5 white and 8 red balls. Two drawings of 3 balls are made such that
i. Balls are replaced before the second trail
j. Balls are not replaced before the second trail
Find the probability that the first drawing will be 3 white and second drawing will be
3 red in each case.
22. How many ‘words’ with five letters are there that strat with vowel and end with an S?
23. A manufacturing firm produces tennis balls in 3 plants with a daily production volume
of 600, 900 and 2000 balls respectively. Account to the past experience it is known that
0.006, 0.007 and 0.010 respectively are defective. If a ball is selected from a days total
production and found to be defective, what is the probability that it comes from the first
plant?
24. First bag contains 3 white, 2 red and 4 black balls, a second bag contains 2 white, 3 red
and 5 black balls and a third bag contains 3 white, 4 red and 2 black balls. Once the bag
is chosen at random and 3 balls are drawn from it. Out of these, two balls are white and
one is red. What are the probabilities that are taken from first, second or third bag?
25. We are given a box containing 5000 VLSI chips, 1000 of which are manufactured by
company X and the rest by company Y. Ten percent of the chips made by company X
are defective and 5% of the chips made by company Y are defective. If a randomly
chosen chip is found to be defective, find the probability that it came from company.
26. If A and B are independent prove that
i)
A and BC are independent,
ii)
AC and B are independent,
iii)
AC and BC are independent
27. There are 3 arrangements of the word DAD, namely DAD, ADD, and DDA. How many
arrangements are there of the word PROBABILITY?
28. Suppose you are taking a multiple-choice test with c choices for each question. In
answering a question on this test, the probability that you know the answer is p. If you
don’t know the answer, you choose one at random. What is the probability that you
knew the answer to a question, given that you answered it correctly?
29. You toss a fair coin three times:
a. What is the probability of three heads, HHH?
b. What is the probability that you observe exactly on heads?
c. Given that you observed at least one heads, what is the probability that you
observe at least two heads?
(Hint : All the coin tosses are independent)
30. In a metropolitan city like bangalore, it’s rainy on third of the days. Knowing this
information, the probability of having heavy traffic is 1/2, and given that it is not rainy,
the probability of having heavy traffic is 1/4. If it’s rainy and there is heavy traffic, an
employee arrive late for work with 1 in 2 chances. On the other fold, the probability of
being late is 1/8 when there its not rainy and there is no heavy traffic. Also in situations
likes rainy and no traffic, not rainy and traffic the probability of being late is 0.25.
Choosing a random day
a. Determine the probability that its not raining and there is a heavy traffic and the
employee is not late to the work
b. Given that an employee arrive late at work, what is the likelihood that it rained
that day?
c. What is the likelihood that an employee is late.
Solve the problem by defining the events as described. Also, tree diagram is
compulsory.
31. The first and second digits of all telephone numbers in a certain area are 7 and 6,
respectively. The third digit can be a 4, 5, or 6. The last four digits can be any number
from 0 to 9. If there here are 18, 243 telephone numbers assigned in the area, how
many are still unassigned?
32. A box contains of 30 red balls and 70 green balls. What is the probability of getting
exactly y red balls in a sample of size 20 if the sampling is done with replacement?
Assume 0≤y≤20.
33. 15 people get on an airport shuttle after their arrival to Bangalore. The airport shuttle
has a route that touches 5 hotels and each person gets off the shuttle at their respective
hotel. The management of airports insisted the drive to collect the information regarding
the number of passengers leave the shuttle at each hotel. How many different
possibilities exist?
34. if a die is rolled 18 times. What is the probability that each number appears exactly 3
times.
35. The 52 cards in a shuffled deck are dealt equally among four players, call them A, B,
C, and D. If A and B have exactly 77spades, what is the probability that C has
exactly 44 spades?
36. The members of a consulting firm rents cars from rental agencies A, B and C as 60%,
30# and 10% respectively. If 9%, 20% and 6% of the cars from A, B and C, respectively,
agencies need tune up and if a rental car delivered to the firm does not need tune up,
what is the probability that it came from B agency.
37. A is know to hit the target in 2 out of 5 shots. Where as B is know to hit the target in 3
out of 4 shots. Find the probability of the target being hit when they both try?
38. The odds in favour of A solving a probability problem are 3 to 4 and the odds against
B solving the problems are 5 to 7. Find the probability that the problem will be solved
by atleast one of them.
39. Ten coins are thrown simultaneously. Find the probability of getting at least 7 heads
40.
41. Two events A and B are such that P(A∩B) = 0.15, P(A U B) =0.65, and P(A|B) = 0.5.
Compute P(B|A).
42. A and B are two independent events defined in the same sample space. They have the
following probabilities P(A) = x and P(B) =y. find the probabilities of the following
events in terms of x and y.
i) Neither event A nor event B occurs
ii) Event A occurs but event B does not occur.
iv)
Either event A occurs or event B does not occur.
v)
43. A problem in statistics is given to 3 students A, B and C whose chances of solving it
are 1/2, 3/4 and 1/4 respectively. What is probability that the problem will be solved.
44. Suppose there are two bags with first bag contains 3 white and 2 black balls, second
bag contains 2 white and 4 black balls. One ball is transferred from bag I to bag II and
then a ball is drawn from the latter. It happens to be white. What is the probability that
the transferred ball is white?
45. Four coins are tossed simultaneously. What is the probability of getting
i) exactly 2 heads
ii) at least 2 heads?
46. A binary communication channel carries data as one of 2 types of signals denoted by 0
and 1. Due to noise, a transmitted 0 is sometimes received as a 1 and a transmitted 1 is
sometimes received as a 0. For a given channel, assume a probability of 0.94 that a
transmitted 0 is correctly received as a 0 and a probability of 0.91 that a transmitted 1
is received as a 1. Further assume a probability of 0.45 of transmitted as 0. If a signal
is sent, determine the probability that
i) a 1 is received
ii) a 1 was transmitted given that a 1 was received.
iii) a 0 was transmitted give that a 0 was received
iv) an error occurred.
47. Three persons X, Y and Z are short-listed for the post of principal of a engineering college.
Their chances of getting selected are in the proportion of 5:3:2 respectively. If X gets the post
the probability of introducing co-education in the college is 0.8. the probabilities of Y and Z
doing the same are 0.6 and 0.4 respectively. What is the probability that co-education will be
introduced in the college?
48. A given lot of IC chips contains 2% defective chips. Each chip is tested before delivery.
The tester itself is not totally reliable. Probability of tester says the chip is good when it really
good is 0.95 and the chip is defective when it actually defective is 0.94. if a tested device is
indicated to be defective, what is the probability that it is actually defective?
49. In a colony, there are 60% men and 40% women. According to a survey taken, 8 men out
of 100 and 10 women out of 100 have high blood pressure. A high B.P person is chosen at
random. What is the probability that this person is male?
50. if A, B, C are mutually independent events, prove that i) A and BUC , ii) A and B∩C are
independent.