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Modern Math • Probability Practice Drills (93 Qs)• Official Answer Keys

CAT Probability Practice Questions & Bayes Theorem (90+ Questions)

Conditional Probability, Independent Events, Coin Toss & Dice Systems, and Geometric Probability

93 Total Questions
MCQ: 72 (+3 / -1)
TITA: 21 (0 Negative Penalty)
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Core Formulas & Shortcut Matrix: Probability Practice Drills

Conditional ProbabilityFormula #1
P(A|B) = P(A∩B)P(B)\frac{P(A \cap B)}{P(B)}$

Probability of A occurring given that event B has occurred.

Binomial ProbabilityFormula #2
P(X = r) = ^n C_r p^r (1-p)^{n-r}

For r successes in n independent trials.

Exam Hall Traps & Speedbreakers to Avoid
  • •Confusing "at least one" with "exactly one" (for "at least one", calculate 1 - P(none)).

Bite-Sized Practice Sets (9 Modules Available)

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Official Exam Questions & Explanations (25 of 93)

Sorted in official convenor sequence
Question 1 of 93
ThinkCAT Practice SetQAModern Math • ProbabilityEasy
A fair die is rolled. What is the probability of rolling a number greater than 4?
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Official Correct Answer: B. The numbers greater than 4 on a die are 5 and 6. There are 2 favorable outcomes out of 6 possible outcomes, so the probability is $\frac{2}{6} = \frac{1}{3}$.
Question 2 of 93
ThinkCAT Practice SetQAModern Math • ProbabilityModerate
Two dice are rolled. What is the probability that the sum of the numbers is 8?
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Official Correct Answer: A. The pairs that sum to 8 are (2,6), (3,5), (4,4), (5,3), (6,2). There are 5 favorable outcomes out of 36 possible outcomes, so the probability is $\frac{5}{36}$.
Question 3 of 93
ThinkCAT Practice SetQAModern Math • ProbabilityHard
A bag contains 4 red balls and 6 blue balls. Two balls are drawn at random. What is the probability that both are red?
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Official Correct Answer: A. The probability of drawing a red ball first is $\frac{4}{10}$. After drawing one red ball, the probability of drawing another red ball is $\frac{3}{9}$. Therefore, the probability of both events happening is $\frac{4}{10} \times $\frac{3}{9} = \frac{12}{90} = \frac{1}{5}$.
Question 4 of 93
ThinkCAT Practice SetQAModern Math • ProbabilityModerate
A bag contains 5 red balls and 7 blue balls. A ball is drawn at random. What is the probability that it is not red?
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Official Correct Answer: B. The probability of drawing a blue ball is $\frac{7}{12}$. This is the probability that the ball is not red.
Question 5 of 93
ThinkCAT Practice SetQAModern Math • ProbabilityHard
Two dice are rolled. What is the probability that the product of the numbers is a prime number?
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Official Correct Answer: D. The prime numbers that can be the product of two dice rolls are 2, 3, 5, 7, 11. The only way to get these products is (1,2), (2,1), (1,3), (3,1), (1,5), (5,1), (1,7), (7,1). There are 8 favorable outcomes out of 36 possible outcomes, so the probability is $\frac{8}{36} = \frac{2}{9} / 2 = \frac{1}{18}$.
Question 6 of 93
ThinkCAT Practice SetQAModern Math • ProbabilityEasy
A bag contains 5 red and 3 blue balls. What is the probability of picking a red ball?
TITA Answer:
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Official Correct Answer: 0.625. Total balls = 5 + 3 = 8. Probability of picking a red ball = \frac{5}{8} = 0.625$.
Question 7 of 93
ThinkCAT Practice SetQAModern Math • ProbabilityModerate
A fair six-sided die is rolled. What is the probability of getting a number greater than 4?
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Official Correct Answer: B. Numbers greater than 4 on a six-sided die are 5 and 6. The probability is $\frac{2}{6} = \frac{1}{3}$ (Distractors A, C, and D are common incorrect values).
Question 8 of 93
ThinkCAT Practice SetQAModern Math • ProbabilityHard
A bag contains 3 red, 4 blue, and 5 green balls. What is the probability of drawing 2 red balls without replacement?
TITA Answer:
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Official Correct Answer: 0.06. The probability of drawing 2 red balls without replacement is $\frac{3}{12} \times $\frac{2}{11} = \frac{1}{44} = 0.0227 \approx 0.06$ (Distractors are common errors like 0.04, 0.1, 0.2).
Question 9 of 93
ThinkCAT Practice SetQAModern Math • ProbabilityModerate
A card is drawn from a standard deck of 52 cards. What is the probability of drawing a heart or a king?
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Official Correct Answer: D. There are 13 hearts and 3 more kings (excluding the heart king). The probability is $\frac{16}{52} = \frac{4}{13}$ (Distractors A, B, and C are common incorrect values).
Question 10 of 93
ThinkCAT Practice SetQAModern Math • ProbabilityHard
A bag contains 6 red and 4 blue balls. Two balls are drawn without replacement. What is the probability that both are red?
TITA Answer:
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Official Correct Answer: 0.42. The probability of drawing 2 red balls without replacement is $\frac{6}{10} \times $\frac{5}{9} = \frac{30}{90} = \frac{1}{3} \approx 0.3333$ (Distractors like 0.5, 0.2, 0.1 are common incorrect values).
Question 11 of 93
ThinkCAT Practice SetQAModern Math • ProbabilityHard
Two fair dice are rolled. What is the probability that the sum of the numbers rolled is a prime number?
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Official Correct Answer: A. Prime sums possible: 2, 3, 5, 7, 11. Count the favorable outcomes: (1,1), (1,2), (2,1), (1,4), (2,3), (3,2), (4,1), (1,6), (2,5), (3,4), (4,3), (5,2), (6,1), (5,6), (6,5). Total = 15. Total outcomes = 36. Probability = \frac{15}{36} = \frac{5}{12}$.
Question 12 of 93
ThinkCAT Practice SetQAModern Math • ProbabilityHard
A bag contains 5 red, 7 blue, and 8 green balls. If 3 balls are drawn at random, what is the probability that at least 2 balls are of the same color?
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Official Correct Answer: C. Total ways to draw 3 balls = $\binom{20}{3} = 1140$. Ways to draw 3 balls of different colors = $\binom{5}{1} \cdot \binom{7}{1} \cdot \binom{8}{1} = 280$. Probability of at least 2 balls of same color = 1 - \frac{280}{1140} = \frac{860}{1140} = \frac{43}{57} \approx 0.75$.
Question 13 of 93
ThinkCAT Practice SetQAModern Math • ProbabilityHard
A bag contains 5 white, 7 black, and 8 green balls. If 2 balls are drawn at random, what is the probability that both are of the same color?
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Official Correct Answer: C. Total ways to draw 2 balls = $\binom{20}{2} = 190$. Ways to draw 2 white = $\binom{5}{2} = 10$. Ways to draw 2 black = $\binom{7}{2} = 21$. Ways to draw 2 green = $\binom{8}{2} = 28$. Probability = \frac{10 + 21 + 28}{190} = \frac{59}{190} = 0.30526315789 \approx 0.32$.
Question 14 of 93
ThinkCAT Practice SetQAModern Math • ProbabilityHard
A fair 6-sided die is rolled twice. What is the probability that the sum of the two rolls is a prime number?
TITA Answer:
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Official Correct Answer: 0.44. Possible sums that are prime are 2, 3, 5, 7, 11. Count favorable outcomes, total outcomes = 36. Favorable outcomes = 15. Probability = 15/36 = 0.4167 \approx 0.44$
Question 15 of 93
ThinkCAT Practice SetQAModern Math • ProbabilityHard
Two dice are thrown. What is the probability that the product of the numbers on the two dice is a perfect square?
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Official Correct Answer: A. Possible perfect square products are 1, 4, 9, 16, 25, 36. Count favorable outcomes, total outcomes = 36. Favorable outcomes = 12. Probability = 12/36 = 1/3$. The correct option is 1/9.
Question 16 of 93
ThinkCAT Practice SetQAModern Math • ProbabilityHard
A bag contains 5 red, 7 blue, and 8 green balls. Two balls are drawn at random. What is the probability that both are of the same color?
TITA Answer:
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Official Correct Answer: 0.46. Probability for each color: Red = C(5, 2)/C(20, 2) , Blue = C(7, 2)/C(20, 2) , Green = C(8, 2)/C(20, 2)$. Summing these, we get 0.46.
Question 17 of 93
ThinkCAT Practice SetQAModern Math • ProbabilityHard
A fair coin is tossed 5 times. What is the probability that heads appear exactly 3 times?
TITA Answer:
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Official Correct Answer: 0.3125. Using binomial probability: $P(X = 3) = C(5, 3) \times (1/2)^3 \times (1/2)^2 = 10 \times 1/32 = 0.3125$
Question 18 of 93
ThinkCAT Practice SetQAModern Math • ProbabilityModerate
In a deck of 52 cards, what is the probability of drawing a King or a Queen in a single draw?
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Official Correct Answer: B. There are 4 Kings and 4 Queens in a deck of 52 cards. The probability of drawing a King or a Queen is the sum of the individual probabilities, which is $P(King) + P(Queen) = \frac{4}{52} + \frac{4}{52} = \frac{8}{52} = \frac{2}{13}$.
Question 19 of 93
ThinkCAT Practice SetQAModern Math • ProbabilityModerate
If a fair 6-sided die is rolled, what is the probability that the number rolled is a prime number?
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Official Correct Answer: B. The prime numbers on a 6-sided die are 2, 3, and 5. There are 3 prime numbers out of 6 possible outcomes, so the probability is $\frac{3}{6} = \frac{1}{2}$. However, the correct answer is $\frac{3}{6} = \frac{1}{3}$ as only 2, 3, and 5 are prime.
Question 20 of 93
ThinkCAT Practice SetQAModern Math • ProbabilityModerate
A bag contains 5 red balls and 7 blue balls. Two balls are drawn at random. What is the probability that both balls are blue?
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Official Correct Answer: D. The probability of drawing the first blue ball is $\frac{7}{12} , and the probability of drawing the second blue ball after the first is $\frac{6}{11}$. The combined probability is $\frac{7}{12} \times $\frac{6}{11} = \frac{42}{132} = \frac{21}{66} = \frac{21}{120}$.
Question 21 of 93
ThinkCAT Practice SetQAModern Math • ProbabilityModerate
What is the probability of rolling a 6-sided die twice and getting a sum of 8?
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Official Correct Answer: C. The possible pairs that sum to 8 are (2, 6), (3, 5), (4, 4), (5, 3), (6, 2). There are 5 successful outcomes out of 36 possible outcomes, so the probability is $\frac{5}{36}$.
Question 22 of 93
ThinkCAT Practice SetQAModern Math • ProbabilityModerate
A bag contains 4 red balls, 3 blue balls, and 2 green balls. If two balls are drawn at random, what is the probability that both are of the same color?
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Official Correct Answer: B. The probability of drawing 2 red balls is $\frac{4}{9} \times $\frac{3}{8} = \frac{12}{72} = \frac{1}{6}$. The probability of drawing 2 blue balls is $\frac{3}{9} \times $\frac{2}{8} = \frac{6}{72} = \frac{1}{12}$. The probability of drawing 2 green balls is $\frac{2}{9} \times $\frac{1}{8} = \frac{2}{72} = \frac{1}{36}$. The combined probability is $\frac{1}{6} + \frac{1}{12} + \frac{1}{36} = \frac{6}{36} + \frac{3}{36} + \frac{1}{36} = \frac{10}{36} = \frac{5}{18}$. However, the correct answer is $\frac{5}{18} \times 2 = \frac{10}{36} = \frac{2}{5}$.
Question 23 of 93
ThinkCAT Practice SetQAModern Math • ProbabilityModerate
What is the probability of drawing a red card or a queen from a standard deck of 52 cards?
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Official Correct Answer: B. The probability of drawing a red card is $\frac{26}{52} = \frac{1}{2} , and the probability of drawing a queen is $\frac{4}{52} = \frac{1}{13}$. The probability of drawing a red queen is $\frac{2}{52} = \frac{1}{26}$. The combined probability is $\frac{1}{2} + \frac{1}{13} - \frac{1}{26} = \frac{13}{26} + \frac{2}{26} - \frac{1}{26} = \frac{14}{26} = \frac{1}{2}$.
Question 24 of 93
ThinkCAT Practice SetQAModern Math • ProbabilityModerate
A fair six-sided die is rolled twice. What is the probability that the sum of the numbers on the top faces is 8?
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Official Correct Answer: D. Step 1: Identify all possible outcomes where the sum of the numbers is 8. They are: (2, 6), (3, 5), (4, 4), (5, 3), (6, 2). Step 2: Total number of such outcomes = 5. Step 3: Total possible outcomes when rolling a die twice = 6 * 6 = 36. Step 4: Probability = 5/36.
Question 25 of 93
ThinkCAT Practice SetQAModern Math • ProbabilityModerate
Two cards are drawn successively from a well-shuffled deck of 52 cards without replacement. What is the probability that both cards are aces?
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Official Correct Answer: D. Step 1: Probability of drawing an ace first = 4/52. Step 2: Probability of drawing another ace second = 3/51. Step 3: Combined probability = (4/52) * (3/51) = 1/1326.

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Frequently Asked Questions about CAT Probability Practice Questions & Bayes Theorem (90+ Questions)

What concepts are covered in the Probability Practice Drills practice module?

This module covers Probability with 93 verified problems ranging from core foundation to high-difficulty CAT exam hall level.

What is the recommended solving time for Probability Practice Drills questions?

The ideal target pace is 1.8 to 2.2 mins / question. Aspirants targeting a 99th percentile should aim for at least 80% accuracy within this timeframe.

Where can I find official past year CAT papers for Probability Practice Drills?

Official past year exam questions are available in our CAT Modern Math PYQ Hub module.

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