Probability Questions

Probability is an important chapter for the students of Class 9, 10, 11, and 12. The Probability Questions, with their answers included in this article, will help you understand the basic concepts and formula. These questions cover concepts like Sample Space, Events, Coin Probability, etc. Solving these problems will improve your understanding and problem-solving skills in probability.

Probability Questions with Solutions

Question 1: A fair six-sided die is rolled. What is the probability of rolling a number greater than 4?

Solution:

There are 2 favorable outcomes (5 and 6) out of 6 possible outcomes.
Probability = Number of favorable outcomes / Total number of outcomes.
= 2 / 6 = 1 / 3 ≈ 0.3333 or 33.33%.

Question 2: A jar contains 3 red, 4 blue, and 5 green marbles. What is the probability of randomly drawing a red marble?

Solution:

Total marbles = 3 (red) + 4 (blue) + 5 (green) = 12.
Probability of drawing a red marble = Number of red marbles / Total number of marbles.
= 3 / 12 = 1 / 4 = 0.25 or 25%.

Question 3: From a deck of 52 cards, what is the probability of drawing an ace or a king?

Solution:

There are 4 aces and 4 kings in the deck.
Total favorable outcomes = 4 (aces) + 4 (kings) = 8.
Probability = Number of favorable outcomes / Total number of outcomes.
= 8 / 52 = 2 / 13 ≈ 0.1538 or 15.38%.

Question 4: A bag contains 10 balls: 6 black and 4 white. Two balls are drawn at random. What is the probability that both are black?

Solution:

Probability of first ball being black = 6/10.
After drawing one black ball, there are 5 black balls left and 9 balls in total.
Probability of second ball being black = 5/9.
Total probability = (6/10) × (5/9) = 30/90 = 1/3 ≈ 0.3333 or 33.33%.

Question 5: What is the probability of getting an even number when a fair six-sided die is rolled?

Solution:

There are 3 even numbers on a die (2, 4, 6).
Total outcomes = 6.
Probability of rolling an even number = Number of even numbers / Total outcomes.
= 3 / 6 = 1 / 2 = 0.5 or 50%.

Question 6: A box contains 2 red, 3 green, and 5 blue pens. If one pen is chosen at random, what is the probability it is not blue?

Solution:

Total pens = 2 (red) + 3 (green) + 5 (blue) = 10.
Number of non-blue pens = 2 (red) + 3 (green) = 5.
Probability of not choosing a blue pen = Number of non-blue pens / Total pens.
= 5 / 10 = 1 / 2 = 0.5 or 50%.

Question 7: What is the probability of flipping a coin three times and getting at least one head?

Solution:

Probability of getting no heads (all tails) = (1/2)³ = 1/8.
Probability of getting at least one head = 1 – Probability of getting no heads.
= 1 – 1/8 = 7/8 = 0.875 or 87.5%.

Question 8: In a class of 20 students, 12 are girls. What is the probability of randomly selecting a boy?

Solution:

Total students = 20.
Number of boys = Total students – Number of girls = 20 – 12 = 8.
Probability of selecting a boy = Number of boys / Total students.
= 8 / 20 = 2 / 5 = 0.4 or 40%.

Question 9: A bag contains 5 red, 7 blue, and 8 green balls. Two balls are drawn without replacement. What is the probability that both balls are red?

Solution:

Probability of first ball being red = 5/20.
After drawing one red ball, there are 4 red balls left and 19 balls in total.
Probability of second ball being red = 4/19.
Total probability = (5/20) × (4/19) = 20/380 = 1/19 ≈ 0.0526 or 5.26%.

Question 10: From a standard deck of 52 cards, what is the probability of drawing a heart or a queen?

Solution:

Total hearts = 13, Total queens = 4.
Since one of the queens is also a heart, there are 3 queens that aren’t hearts.
Total favorable outcomes = 13 (hearts) + 3 (other queens) = 16.
Probability = Number of favorable outcomes / Total outcomes.
= 16 / 52 = 4 / 13 ≈ 0.3077 or 30.77%.

Related Links:

• Probabiliy in Maths
• Events in Probability
• Coin Toss Probability

Question 11: A bag contains 4 red, 6 blue, and 5 green marbles. If two marbles are drawn randomly, what is the probability that both are the same color?

Solution:

Total marbles = 4 (red) + 6 (blue) + 5 (green) = 15.
Probability of drawing 2 red marbles = (4/15) × (3/14).
Probability of drawing 2 blue marbles = (6/15) × (5/14).
Probability of drawing 2 green marbles = (5/15) × (4/14).
Total probability = Sum of the above probabilities.
= (4/15) × (3/14) + (6/15) × (5/14) + (5/15) × (4/14).

Question 12: A spinner with numbers 1 to 5 is spun twice. What is the probability that the sum of the two spins is 6?

Solution:

The possible outcomes that sum to 6 are: (1,5), (2,4), (3,3), (4,2), and (5,1).
Total outcomes when spinning twice = 5 × 5 = 25.
Probability = Number of favorable outcomes / Total outcomes.
= 5 / 25 = 1 / 5 = 0.2 or 20%.

Question 13: What is the probability of not drawing a face card (Jack, Queen, King) from a standard deck of 52 cards?

Solution:

Total face cards in the deck = 12 (4 Jacks, 4 Queens, 4 Kings).
Total non-face cards = 52 – 12 = 40.
Probability of not drawing a face card = Number of non-face cards / Total cards.
= 40 / 52 = 10 / 13 ≈ 0.7692 or 76.92%.

Question 14: A jar contains 10 balls, of which 3 are red and 7 are black. If one ball is drawn and then replaced, and a second ball is drawn, what is the probability that both balls are red?

Solution:

Probability of drawing a red ball and replacing it = 3/10.
Since the ball is replaced, the probability of drawing a red ball again = 3/10.
Total probability = (3/10) × (3/10) = 9/100 = 0.09 or 9%.

Question 15: In a lottery, there are 100 tickets numbered from 1 to 100. What is the probability that a ticket drawn at random has a number which is a multiple of 5 or 7?

Solution:

Number of multiples of 5 between 1 and 100 = 20.
Number of multiples of 7 between 1 and 100 = 14.
Number of multiples of both 5 and 7 (35 and 70) = 2.
Total favorable outcomes = 20 + 14 – 2 = 32.
Probability = Number of favorable outcomes / Total tickets.
= 32 / 100 = 0.32 or 32%.

Question 16: A box contains 5 red, 3 green, and 2 blue balls. Two balls are drawn at random. What is the probability that they are of different colors?

Solution:

Total balls = 5 (red) + 3 (green) + 2 (blue) = 10.
Probability of drawing one red and one green = (5/10) × (3/9) + (3/10) × (5/9).
Probability of drawing one red and one blue = (5/10) × (2/9) + (2/10) × (5/9).
Probability of drawing one green and one blue = (3/10) × (2/9) + (2/10) × (3/9).
Total probability = Sum of the above probabilities.

Question 17: In a class of 60 students, 30 are boys. If two students are selected at random, what is the probability that both are girls?

Solution:

Total girls = 60 – 30 = 30.
Probability of first student being a girl = 30/60.
After selecting one girl, remaining girls = 29, total students = 59.
Probability of second student being a girl = 29/59.
Total probability = (30/60) × (29/59).

Question 18: A fair die is rolled twice. What is the probability that the sum of the two rolls is 8?

Solution:

The pairs that sum to 8 are: (2,6), (3,5), (4,4), (5,3), (6,2).
Total outcomes when rolling twice = 6 × 6 = 36.
Probability = Number of favorable outcomes / Total outcomes.
= 5 / 36 ≈ 0.1389 or 13.89%.

Question 19: A bag contains 15 balls numbered 1 to 15. What is the probability of drawing a number that is either even or a multiple of 3?

Solution:

Even numbers (2, 4, 6, 8, 10, 12, 14) = 7.
Multiples of 3 (3, 6, 9, 12, 15) = 5.
Common numbers in both sets (6, 12) = 2.
Total favorable outcomes = 7 (even) + 5 (multiples of 3) – 2 (common) = 10.
Probability = Number of favorable outcomes / Total balls.
= 10 / 15 = 2 / 3 ≈ 0.6667 or 66.67%.

Question 20: A fair coin is tossed 5 times. What is the probability of getting exactly 3 heads?

Solution:

The number of ways to get exactly 3 heads in 5 tosses can be calculated using the binomial formula: C(n, k) = n! / [k!(n-k)!], where C is the combination, n is the total number of events, and k is the number of successful events.
C(5, 3) = 5! / [3!(5-3)!] = 10.
Probability of getting a head in one toss = 1/2.
Probability = 10 × (1/2)³ × (1/2)² = 10/32 = 0.3125 or 31.25%.

Question 21: A deck of 52 cards is shuffled. What is the probability of drawing either a heart or a queen, but not the queen of hearts?

Solution:

Total hearts = 13, including the queen of hearts.
Total queens = 4, including the queen of hearts.
Favorable outcomes = (13 hearts – 1 queen of hearts) + (4 queens – 1 queen of hearts) = 12 + 3 = 15.
Probability = Number of favorable outcomes / Total outcomes = 15 / 52 ≈ 0.2885 or 28.85%.

Question 22: A bag contains 5 red, 6 blue, and 7 green balls. Three balls are drawn without replacement. What is the probability that they are drawn in the order red, blue, and green?

Solution:

Probability of drawing a red ball first = 5/18.
Then, probability of drawing a blue ball = 6/17 (after removing one red ball).
Finally, probability of drawing a green ball = 7/16 (after removing one red and one blue ball).
Total probability = (5/18) × (6/17) × (7/16) ≈ 0.0456 or 4.56%.

Question 23: In a lottery, there are 100 tickets numbered 1 to 100. What is the probability of drawing a number that is both a multiple of 5 and a multiple of 7?

Solution:

Multiples of 5 and 7 between 1 and 100 are 35 and 70.
Total favorable outcomes = 2.
Probability = Number of favorable outcomes / Total tickets = 2 / 100 = 0.02 or 2%.

Question 24: A fair die is rolled three times. What is the probability of rolling a 6 each time?

Solution:

The probability of rolling a 6 on one roll = 1/6.
The probability of rolling three 6s in three rolls = (1/6) × (1/6) × (1/6) = 1/216 ≈ 0.00463 or 0.463%.

Question 25: A test consists of 10 true/false questions. To pass, a student must answer at least 8 questions correctly. If a student guesses on each question, what is the probability of passing the test?

Solution:

The probability of guessing each question correctly = 1/2.
The probability of getting exactly 8, 9, or 10 questions right can be calculated using the binomial probability formula: P(X=k) = C(n, k) × (p)^k × (1-p)^n-k, where C(n, k) is the combination.
Calculate P(8) + P(9) + P(10) using the formula with n = 10, k = 8, 9, 10, and p = 1/2.
Sum these probabilities to find the total probability of passing.

Conclusion

Through this article, we’ve explored various probability questions with solutions, offering you a good resource to practise and refine your skills. From basic probability concepts to more complex problems, these questions, based on the NCERT guidelines cater to learners at all levels.