Probability - Key Formulas
Basic Probability
Probability of an event = Number of favorable outcomes / Total number of possible outcomes
Complement Rule
Probability of event not occurring = 1 - Probability of event occurring
Addition Rule
Union of Events: P(A or B) = P(A) + P(B) - P(A and B)
Mutually Exclusive Events: P(A or B) = P(A) + P(B)
Multiplication Rule
Independent Events: P(A and B) = P(A) × P(B)
Dependent Events: P(A and B) = P(A) × P(B|A)
Probability Problems
Practice aptitude questions on probability concepts and applications
Question 1:
Which of the following options is correct with respect to the probability of an event A?
0 ≤ P(A) ≤ 1 is correct because the probability of any event is always between more than or equal to zero and less than or equal to one.
Question 2:
Which of the following options is incorrect with respect to an event A?
P(A) > 1 is incorrect as the probability of an event can never exceed one, it can acquire a maximum value of one.
Question 3:
What are the events called which can never occur together according to the probability theory?
Mutually exclusive events are things that can't happen at the same time. For example, you can't get head and tail at the same time while tossing a single coin.
Question 4:
What is the term used for all possible outcomes for a random experiment?
Sample space is the set of all possible outcomes of a random experiment. For example, when tossing a coin, the sample space is {H, T}.
Question 5:
Which of the following terms explains the probability of an event irrespective of the outcome of another variable?
Marginal probability is correct as Marginal probability is the term used to describe the probability of an event irrespective of the outcome of another variable.
Question 6:
What will be the marginal probability of dependent and independent events?
Marginal probability is the probability of an event irrespective of the outcome of another variable. Thus, the marginal probability of dependent and independent events will be same.
Question 7:
What is the probability of non – occurrence of an event Q?
Considering an event Q the probability of non – occurrence of Q is complement of Q as both would sum up equal to one. Probability of Q + probability of non – occurrence of Q = 1. Thus, probability of non – occurrence of Q is known as complement of Q.
Question 8:
What will be the joint probability of two independent events M and N?
Given, two independent events M and N. Joint probability = P(M) * P(N).
Question 9:
Which of the following is a condition for binomial distribution in probability?
By definition the basic condition for binomial distribution in probability is that it must have only two possible outcomes. Thus, only two possible outcomes is the correct answer.
Question 10:
In which of the following events the rule of addition of probability is used?
In the case of mutually exclusive events the use of addition of probability rule is allowed to calculate the probability of the events occurring together.
Question 11:
What is the probability that a number selected at random from 1 to 15 is a multiple of 3?
Total possible outcomes = 15
Favorable outcomes = 3, 6, 9, 12, 15
Probability = 5/15 = 1/3
Question 12:
If the probability of an event is Q, then what will be the probability of its complementary event?
Given, Probability = Q
Probability of complementary of Q = 1 – Q.
Question 13:
What is the probability of having 53 Tuesdays in a non – leap year?
A non – Leap year has 365 days which is 52 weeks and 1 extra day.
For this one day
Total possible outcomes = 7 (from Sunday to Saturday)
Favorable outcomes = 1 (Tuesday)
Probability = 1/7
Question 14:
Hundred balls are numbered from 1 to 100. What is the probability of getting a ball having a prime number?
Total possible outcomes = 100
Favorable outcomes = (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97) = 25
Thus, probability = 25/100 = 1/4
Question 15:
A bag contains 10 red cards and some yellow cards. If the probability of getting yellow cards is thrice the probability of getting a red card, then what is the number of yellow cards?
Given, Red cards = 10
Let the yellow cards be x. Then total number of cards = 10 + x.
According to the question
x/(10 + x) = 3 * 10/(10 + x)
➩ x = 30
Question 16:
A box contains 1000 pens of which 25 are defective. What is the probability of drawing a defective pen from the box?
Total possible outcomes = 1000
Favorable outcomes = 25
Probability = 25/1000 = 1/40
Question 17:
Cards marked with numbers 1 to 100 are placed in a box. One card is drawn from this box randomly, then what is the probability that the number on the card is a perfect square?
Total possible outcomes = 100
Favorable outcomes = (1, 4, 9, 16, 25, 36, 49, 64, 81, 100) = 10
Probability = 10/100 = 1/10
Question 18:
A number is selected at random from the first fifty natural numbers. What is the probability that the number is divisible by 2, 3 and 4?
Total possible outcomes = 50
Favorable outcomes = 12, 24, 36, 48 = 4
Probability = 4/50 = 2/25
Question 19:
A jar has 48 marbles of which few are black and others are red. If the probability of getting a black marble is 3/4, then what will be the number of red marbles?
Let the number of red marbles be z.
P(red) = 1 – P(black)
➩ z/48 = 1 – 3/4
➩ z/48 = 1/4
➩ z = 48/4
➩ z = 12
Question 20:
What is the probability of getting 53 Wednesdays in a leap year?
Leap year has 52 weeks and two days i.e., 52 Wednesdays are there and two days are left.
Those two days can be = (Sunday, Monday), (Monday, Tuesday), (Tuesday, Wednesday), (Wednesday, Thursday), (Thursday, Friday), (Friday, Saturday), (Saturday, Sunday)
There are two pairs having Wednesday.
Thus, Probability (53 Wednesday) = 2/7.
Question 21:
What is the sum of the probability of an event and its complementary event occurring together?
Complementary of an event is also known as non – event or event not occurring. So, the sum of the probability of an event and non – event occurring is always equal to one.
Question 22:
In which type of events the rule of multiplication of probability is applicable?
The rule of multiplication of probability is applicable in independent events which are not related to find their joint probability.
Question 23:
Which of the following is not an example of discrete probability distribution?
The weight of a person is not a discrete value as it can go on to a number of places after the decimal.
Question 24:
Which of the following is a correct option regarding the Poisson distribution?
The mean and variance in Poisson distribution are always the same. If A is the average number of successful outcomes in a Poisson distribution, then the mean and variance of the distribution will become equal to A. As in Poisson distribution only single parameter is required for estimating the probability.
Question 25:
Which of the following is not a condition of the binomial distribution?
There is no condition like trials must be at least 4 for binomial distribution. The only conditions for a binomial distribution is that the number of outcomes of a particular event must be only two like in a coin its only either a head or a tail and nothing else.
Question 26:
The probability of an event occurring is 0.39, then what will be the probability of this event not occurring?
Given, P(A) = 0.39
P(not A) = 1 – 0.39 = 0.61
Question 27:
What are the total number of outcomes when five coins are tossed together?
Given, n = 5
Total number of outcomes = 2^5 = 32
Question 28:
Which of the following terms relates to the sum total of all the outcomes in probability?
Exhaustive event means the sum total of all the outcomes in probability. For example, when we roll a die once then the collection of all the possible events i.e., 1, 2, 3, 4, 5, 6 are known as exhaustive events.
Question 29:
A bag contains 3 white balls and 2 black balls. What will be the probability of drawing a red ball when drawn at random?
Since there is no red ball in the bag, the probability of drawing a red ball is 0.
Question 30:
The probability that tomorrow will be a sunny day is 0.68, then what is the probability that tomorrow will not be a sunny day?
Given, P(Sunny day) = 0.68
P(Not sunny day) = 1 – P(Sunny day) = 1 – 0.68 = 0.32
Question 31:
What are the total number of outcomes when three coins are tossed together?
Given, n = 3
Total possible outcomes = 2^n = 2^3 = 8
Question 32:
What is the sum of the probability of an event and non-event?
The sum of the probability of an event and non-event is always 1.
Question 33:
An alphabet is chosen at random from the word "MATHEMATICS". What is the probability of it being a vowel?
Total number of possible outcomes = 11
Favorable outcomes = 4
Probability = 4/11
Question 34:
What is the probability of getting a number that is divisible by 5 from 1 to 15?
Total possible outcomes = 15
Favorable outcomes = 3
Probability = 3/15 = 1/5
Question 35:
The following probabilities are given with respect to an event. Which of the following is wrong?
8/5 is wrong as probability of any event can never be greater than 1 or less than 0.
Question 36:
What are the total number of outcomes when 5 dices are rolled together?
Given, n = 5
Total number of possible outcomes = 6^n = 6^5 = 7776
Question 37:
A bag contains 10 black balls, 5 red balls and 5 white balls. What is the probability of getting either a red or a white ball?
Total number of possible outcomes = 20
Favorable outcomes = 10
Probability = 10/20 = 1/2
Question 38:
A dice is rolled. What is the probability that we get a number divisible by 3?
Total possible outcomes = 6
Favorable outcomes = 3, 6 = 2
Probability = 2/6 = 1/3
Question 39:
Which of the following is a classic example of binomial distribution of probability?
Toss a coin is a classic example of binomial distribution of probability as there are only two outcomes in this case namely head or tail. Also, both the outcomes have an equal chance of coming which remains constant for any number of trials. This constant probability of getting a head or a tail is 1/2.
Question 40:
When you toss a coin twice, what is the probability of getting a head in both the attempts?
Possible outcomes = HH, TT, HT, TH
Favorable outcomes = HH
Thus, Probability = 1/4.
Question 41:
When you toss a coin what is the probability of not getting a head?
P(Not Head) = 1 – P(H) = 1 – 1/2 = 1/2
Question 42:
When you toss a coin twice what is the probability of getting a tail in both the attempts?
Possible outcomes = HH, TT, HT, TH
Favorable outcomes = TT
Thus, Probability = 1/4.
Question 43:
Two unbiased coins are tossed. What are the total number of outcomes?
Possible outcomes = (HH, HT, TH, TT)
Thus, total number of outcomes are 4.
Question 44:
Two coins are tossed simultaneously. What is the probability of getting only one head after taking all the possibilities into consideration?
Possible outcomes = (HH, HT, TH, TT)
Favorable outcomes = HT, TH
Thus, probability = 1/2.
Question 45:
Two unbiased coins are tossed. What is the probability of getting no less than 1 head after taking all the possibilities into consideration?
Possible outcomes = (HH, HT, TH, TT)
Favorable outcomes = HH, HT, TH
Thus, probability = 3/4.
Question 46:
Two unbiased coins are tossed. What is the probability of getting only one tail after taking all the possibilities into consideration?
Possible outcomes = (HH, HT, TH, TT)
Favorable outcomes = HT, TH
Thus, probability = 1/2.
Question 47:
Two unbiased coins are tossed. What is the probability of getting two tail?
Possible outcomes = (HH, HT, TH, TT)
Favorable outcomes = TT
Thus, probability = 1/4.
Question 48:
Which of the following options is incorrect when one coin is tossed?
P(H) * P(T) = 1 is incorrect as,
P(H) * P(T) = 1/2 * 1/2 = 1/4
Question 49:
Three unbiased coins are tossed. What are the total number of outcomes possible?
Possible outcomes = (HHH, HHT, HTH, THH, THT, HTT, TTH, TTT)
Thus, total number of outcomes are 8.
Question 50:
What is the correct expression for finding the total number of outcomes when n coins are tossed at a time?
Given, Number of coins = n.
Total number of possible outcomes = 2^n.
Question 51:
Three unbiased coins are tossed. What is the probability of getting only one H?
Given, coins = 3
Possible outcomes = (HHH, HHT, HTH, THH, THT, HTT, TTH, TTT)
Favorable outcomes = (THT, HTT, TTH)
Thus, probability = 3/8.
Question 52:
Three unbiased coins are tossed. What is the probability of getting at most one H?
Given, coins = 3 Possible outcomes = (HHH, HHT, HTH, THH, THT, HTT, TTH, TTT) Favorable outcomes = (THT, HTT, TTH, TTT) Thus, probability = 4/8 = 1/2.
Question 53:
Three unbiased coins are tossed. What is the probability of getting at most two tail?
Given, coins = 3 Possible outcomes = (HHH, HHT, HTH, THH, THT, HTT, TTH, TTT) Favorable outcomes = (THT, HTT, TTH, HHH, HHT, HTH, THH) Thus, probability = 7/8.
Question 54:
Three unbiased coins are tossed. What is the probability of getting three tail?
Given, coins = 3 Possible outcomes = (HHH, HHT, HTH, THH, THT, HTT, TTH, TTT) Favorable outcomes = (TTT) Thus, probability = 1/8.
Question 55:
Three unbiased coins are tossed. What is the probability of getting minimum two head?
Given, coins = 3 Possible outcomes = (HHH, HHT, HTH, THH, THT, HTT, TTH, TTT) Favorable outcomes = (HHT, HTH, THH, HHH) Thus, probability = 4/8 = 1/2.
Question 56:
Three unbiased coins are tossed. What is the probability of getting same event in all the three throws?
Given, coins = 3 Possible outcomes = (HHH, HHT, HTH, THH, THT, HTT, TTH, TTT) Favorable outcomes = (HHH, TTT) Thus, probability = 2/8 = 1/4.
Question 57:
Ten unbiased coins are tossed. What are the total number of outcomes possible?
Given, n = 10 Total number of outcomes = 2^n = 2^10 = 1024
Question 58:
Three unbiased coins are tossed. What is the probability of getting complement of having a tail?
Given, coins = 3 Possible outcomes = (HHH, HHT, HTH, THH, THT, HTT, TTH, TTT) Favorable outcomes = (HHH) Thus, probability = 1/8.
Question 59:
An unbiased dice is thrown. What are the total number of possible outcomes?
Possible outcomes = 1, 2, 3, 4, 5, 6 Thus, total number of possible outcomes when an unbiased dice is thrown are 6.
Question 60:
An unbiased dice is thrown. What is the probability of getting 3?
Possible outcomes = 1, 2, 3, 4, 5, 6 Favorable outcome = 3 Probability = 1/6
Question 61:
An unbiased dice is thrown. What is the probability of getting 10?
Possible outcomes = 1, 2, 3, 4, 5, 6 Favorable outcome = 0 Probability = 0/6 = 0
Question 62:
An unbiased dice is thrown. What is the probability of getting a number greater than and equal to 3?
Possible outcomes = 1, 2, 3, 4, 5, 6 Favorable outcome = 3, 4, 5, 6 Probability = 4/6 = 2/3
Question 63:
An unbiased dice is thrown. What is the probability of getting a number less than 3?
Possible outcomes = 1, 2, 3, 4, 5, 6 Favorable outcome = 1, 2 Probability = 2/6 = 1/3
Question 64:
An unbiased dice is thrown. What is the probability of getting an odd number?
Possible outcomes = 1, 2, 3, 4, 5, 6 Favorable outcome = 1, 3, 5 Probability = 3/6 = 1/2
Question 65:
An unbiased dice is thrown. What is the probability of getting 5 or a greater number?
Possible outcomes = 1, 2, 3, 4, 5, 6 Favorable outcome = 5, 6 Probability = 2/6 = 1/3
Question 66:
An unbiased dice is thrown. What is the probability of getting an even number greater than 3?
Possible outcomes = 1, 2, 3, 4, 5, 6 Favorable outcome = 4, 6 Probability = 2/6
Question 67:
An unbiased dice is thrown. What is the probability of getting an even number greater than 0?
Possible outcomes = 1, 2, 3, 4, 5, 6 Favorable outcome = 2, 4, 6 Probability = 3/6 = 1/2
Question 68:
n dice are thrown simultaneously. What is the total number of possible outcomes?
Given, Number of dice = n Total number of possible outcomes = 6^n as according to the theory of combination we can get 6 different outcomes in throw of a single dice and since n dice are thrown simultaneously. Thus, total possible outcomes will be represented by 6^n.
Question 69:
Two dice are thrown simultaneously. What will be the total number of possible outcomes?
Given, Number of dice = 2 Total number of possible outcomes = 6^n = 6^2 = 36
Question 70:
Two dice are thrown simultaneously. What is the probability of getting equal digits in both the dice?
Total number of possible outcomes = 36 Favorable outcomes = (11, 22, 33, 44, 55, 66) = 6 Probability = 6/36 = 1/6
Question 71:
Two dice are thrown simultaneously. What is the probability of getting a number not divisible by two in both the dice?
Total number of possible outcomes = 36 Favorable outcomes = (11, 13, 15, 31, 33, 35, 51, 53, 55) = 9 Probability = 9/36 = 1/4
Question 72:
Two dice are thrown simultaneously. What is the probability of getting a number greater than 4 in both the dice?
Total number of possible outcomes = 36 Favorable outcomes = (55, 56, 65, 66) = 4 Probability = 4/36 = 1/9
Question 73:
Two dice are thrown simultaneously. What is the probability of getting a prime number in both the dice?
Total number of possible outcomes = 36 Favorable outcomes = (22, 23, 25, 32, 33, 35, 52, 53, 55) = 9 Probability = 9/36 = 1/4
Question 74:
Two dice are thrown simultaneously. What is the probability of getting a odd number greater than 4 in both the dice?
Total number of possible outcomes = 36 Favorable outcomes = 5,5 = 1 Probability = 1/36
Question 75:
Two dice are thrown simultaneously. What is the probability of getting sum of both the numbers as 9?
Total number of possible outcomes = 36 Favorable outcomes = (36, 45, 54, 63) = 4 Probability = 4/36 = 1/9
Question 76:
Two dice are thrown simultaneously. What is the probability of getting difference of both the numbers as 4?
Total number of possible outcomes = 36 Favorable outcomes = (15, 26, 51, 62) Probability = 4/36 = 1/9
Question 77:
Two dice are thrown simultaneously. What is the probability of getting a number divisible by 2 on the first dice and a number divisible by 3 on the second dice simultaneously?
Total number of possible outcomes = 36 Favorable outcomes = (23, 26, 43, 46, 63, 66) Probability = 6/36 = 1/6
Question 78:
Three dice are thrown simultaneously. What are the total number of possible outcomes?
Given, Number of dice = 3 Total number of possible outcomes = 6^3 = 216
Question 79:
What is the probability of getting a club from a well shuffled deck of 52 cards?
Total possible outcomes = 52 Favorable outcomes = 13 Probability = 13/52 = 1/4
Question 80:
What will be the probability of getting a jack of the spade from a well shuffled deck of 52 cards?
Total possible outcomes = 52 Favorable outcomes = 1 Probability = 1/52
Question 81:
What will be the probability of getting a queen from a well shuffled deck of 52 cards?
Total possible outcomes = 52 Favorable outcomes = 4 Probability = 4/52 = 1/13
Question 82:
What will be the probability of getting a red card from a well shuffled deck of 52 cards?
Total possible outcomes = 52 Favorable outcomes = 26 Probability = 26/52 = 1/2
Question 83:
What is the probability of getting a face card from a well shuffled deck of 52 cards?
Total possible outcomes = 52 Favorable outcomes = 12 Probability = 12/52 = 3/13
Question 84:
What is the probability of getting a numbered card from a well shuffled deck of 52 cards?
Total possible outcomes = 52 Favorable outcomes = 36 Probability = 36/52 = 9/13
Question 85:
What is the probability of getting a black ace from a well shuffled deck of 52 cards?
Total possible outcomes = 52 Favorable outcomes = 2 Probability = 2/52 = 1/26
Question 86:
What is the probability of getting a number 8 from a well shuffled deck of 52 cards?
Total possible outcomes = 52 Favorable outcomes = 4 Probability = 4/52 = 1/13
Question 87:
What is the probability of getting a red numbered card greater than 6 from a well shuffled deck of 52 cards?
Total possible outcomes = 52 Favorable outcomes = 8 Probability = 8/52 = 2/13
Question 88:
What is the probability of getting a non – numbered card from a well shuffled standard deck?
Total possible outcomes = 52 Favorable outcomes = 16 Probability = 16/52 = 4/13
Question 89:
What is the probability of drawing two kings from a well shuffled pack of cards without replacement?
Total cards = 52 Probability of 1st card being a king = 4/52 = 1/13 Probability of 2nd card being a king = 3/51 = 1/17 Total probability = 1/13 * 1/17 = 1/221
Question 90:
What is the probability of drawing two spades from a well shuffled pack of cards without replacement?
Total cards = 52 Probability of 1st card being a spade = 13/52 = 1/4 Probability of 2nd card being a spade = 12/51 = 4/17 Total probability = 1/4 * 4/17 = 1/17
Question 91:
What is the probability of drawing one spade and one club from a well shuffled pack of cards without replacement?
Total cards = 52 Probability of 1st card being a spade = 13/52 = 1/4 Probability of 2nd card being a club = 13/51 Total probability = 1/4 * 13/51 = 13/204
Question 92:
What is the probability of drawing two numbered cards from a well shuffled pack of cards without replacement?
Total cards = 52 Probability of 1st card being a number = 36/52 = 9/13 Probability of 2nd card being a number = 35/51 Total probability = 9/13 * 35/51 = 315/663 = 105/221
Question 93:
What is the probability of drawing two queens of diamond from a well shuffled pack of cards without replacement?
Total cards = 52 Probability of 1st card being a diamond queen = 1/52 Probability of 2nd card being a diamond queen = 0/51 = 0 Total probability = 1/52 * 0 = 0
Question 94:
What is the probability of getting two ace cards from a well shuffled standard deck with replacement?
Total cards = 52 Probability of 1st card being an ace = 4/52 = 1/13 Probability of 2nd card being an ace = 4/52 = 1/13 Total probability = 1/13 * 1/13 = 1/169
Question 95:
What is the probability of getting two heart cards from a well shuffled standard deck with replacement?
Total cards = 52 Probability of 1st card being a heart = 13/52 = 1/4 Probability of 2nd card being a heart = 13/52 = 1/4 Total probability = 1/4 * 1/4 = 1/16
Question 96:
What is the probability of getting two numbered cards from a well shuffled standard deck with replacement?
Total cards = 52 Probability of 1st card being a number = 36/52 = 9/13 Probability of 2nd card being a number = 36/52 = 9/13 Total probability = 9/13 * 9/13 = 81/169
Question 97:
What is the probability of getting two non-numbered cards from a well shuffled standard deck with replacement?
Total cards = 52 Probability of 1st card being a non-number = 16/52 = 4/13 Probability of 2nd card being a non-number = 16/52 = 4/13 Total probability = 4/13 * 4/13 = 16/169
Question 98:
What is the probability of getting two black cards from a well shuffled standard deck with replacement?
Total cards = 52 Probability of 1st card being a black card = 26/52 = 1/2 Probability of 2nd card being a black card = 26/52 = 1/2 Total probability = 1/2 * 1/2 = 1/4