Extra Practice Questions
Probability (R.D. Sharma Level)
1. A card is drawn at random from a well-shuffled deck of 52 cards. Find the probability that the card drawn is (i) a face card or a spade (ii) neither a king nor a red queen.
Total cards = 52.
(i) a face card or a spade:
Number of face cards = 12. Number of spades = 13.
Number of cards that are both a face card and a spade (J, Q, K of spades) = 3.
P(A or B) = P(A) + P(B) – P(A and B)
P(Face or Spade) = 12/52 + 13/52 – 3/52 = 22/52 = 11/26.
(ii) neither a king nor a red queen:
Number of kings = 4. Number of red queens = 2 (Queen of hearts, Queen of diamonds).
Total cards to be excluded = 4 + 2 = 6.
Number of favourable cards = 52 – 6 = 46.
P(neither king nor red queen) = 46/52 = 23/26.
2. A bag contains cards numbered from 1 to 49. A card is drawn at random. Find the probability that the number on the card is (i) a prime number (ii) a multiple of 5 (iii) a perfect square.
Total cards = 49.
(i) a prime number:
Prime numbers up to 49: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.
Total favourable outcomes = 15. P(prime) = 15/49.
(ii) a multiple of 5:
Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, 45.
Total favourable outcomes = 9. P(multiple of 5) = 9/49.
(iii) a perfect square:
Perfect squares: 1, 4, 9, 16, 25, 36, 49.
Total favourable outcomes = 7. P(perfect square) = 7/49 = 1/7.
3. In a single throw of a pair of different dice, what is the probability of getting (i) a prime number on each die? (ii) a total of 9 or 11?
Total outcomes = 6 × 6 = 36.
(i) a prime number on each die:
Prime numbers on a die are 2, 3, 5.
Favourable pairs: (2,2), (2,3), (2,5), (3,2), (3,3), (3,5), (5,2), (5,3), (5,5). Total = 9.
P(prime on each) = 9/36 = 1/4.
(ii) a total of 9 or 11:
Outcomes for sum 9: (3,6), (4,5), (5,4), (6,3). (4 outcomes)
Outcomes for sum 11: (5,6), (6,5). (2 outcomes)
Total favourable outcomes = 4 + 2 = 6.
P(sum is 9 or 11) = 6/36 = 1/6.
4. A bag contains 18 balls out of which x balls are red. (i) If one ball is drawn at random, what is the probability that it is red? (ii) If 2 more red balls are put in the bag, the probability of drawing a red ball will be 9/8 times the probability of drawing a red ball in the first case. Find the value of x.
(i) Total balls = 18. Red balls = x. P(red) = x/18.
(ii) New total balls = 18 + 2 = 20. New red balls = x + 2.
New P(red) = (x+2)/20.
Given: New P(red) = (9/8) × Old P(red).
(x+2)/20 = (9/8) × (x/18)
(x+2)/20 = 9x / 144 = x / 16.
16(x+2) = 20x
16x + 32 = 20x
4x = 32 => x = 8.
Answer: The value of x is 8.
5. What is the probability that a leap year, selected at random, will contain 53 Sundays?
A leap year has 366 days.
366 days = 52 weeks and 2 extra days. (366 = 52 × 7 + 2).
The 52 weeks will contain 52 Sundays for sure. The 53rd Sunday depends on the remaining 2 extra days.
The possible combinations for these 2 extra days are:
{ (Sun, Mon), (Mon, Tue), (Tue, Wed), (Wed, Thu), (Thu, Fri), (Fri, Sat), (Sat, Sun) }
Total possible outcomes = 7.
The favourable outcomes (where one of the days is a Sunday) are (Sun, Mon) and (Sat, Sun). Count = 2.
Probability = Favourable / Total = 2/7.
Answer: The probability is 2/7.
6. A number x is selected from the numbers 1, 2, 3 and then a second number y is randomly selected from the numbers 1, 4, 9. What is the probability that the product xy of the two numbers will be less than 9?
The possible pairs of (x, y) are:
(1,1), (1,4), (1,9)
(2,1), (2,4), (2,9)
(3,1), (3,4), (3,9)
Total number of outcomes = 3 × 3 = 9.
Now, we find the product xy for each pair and check if it’s less than 9:
1×1=1 (<9), 1×4=4 (<9), 1×9=9 (not <9)
2×1=2 (<9), 2×4=8 (<9), 2×9=18 (not <9)
3×1=3 (<9), 3×4=12 (not <9), 3×9=27 (not <9)
The favourable outcomes correspond to the products {1, 4, 2, 8, 3}. There are 5 such outcomes.
P(xy < 9) = Favourable / Total = 5/9.
Answer: The probability is 5/9.
7. A letter is chosen at random from the letters of the word ‘ASSASSINATION’. Find the probability that the letter chosen is (i) a vowel (ii) a consonant.
Total letters = 13.
(i) a vowel: The vowels are A, A, I, A, I, O. Total = 6.
P(vowel) = 6/13.
(ii) a consonant: The consonants are S, S, S, S, N, T, N. Total = 7.
P(consonant) = 7/13. (or 1 – 6/13).
Answer: P(vowel) = 6/13, P(consonant) = 7/13.
8. All the black face cards are removed from a pack of 52 playing cards. The remaining cards are well shuffled and then a card is drawn at random. Find the probability of getting a (i) face card, (ii) red card, (iii) black card.
Black face cards (J,Q,K of spades and clubs) = 6 cards.
Remaining cards = 52 – 6 = 46.
(i) face card:
The 6 red face cards are still in the pack. Favourable = 6.
P(face card) = 6/46 = 3/23.
(ii) red card:
All 26 red cards are still in the pack. Favourable = 26.
P(red card) = 26/46 = 13/23.
(iii) black card:
The 26 black cards minus the 6 black face cards = 20 black cards remain.
P(black card) = 20/46 = 10/23.
9. A number is chosen at random from the integers -3, -2, -1, 0, 1, 2, 3. What is the probability that the square of this number is less than or equal to 1?
Total integers = 7.
Let’s find the square of each number:
(-3)²=9, (-2)²=4, (-1)²=1, (0)²=0, (1)²=1, (2)²=4, (3)²=9.
We need the square to be less than or equal to 1. The favourable integers are -1, 0, 1.
Number of favourable outcomes = 3.
Probability = Favourable / Total = 3/7.
Answer: The probability is 3/7.
10. Two customers are visiting a particular shop in the same week (Tuesday to Saturday). Each is equally likely to visit the shop on any one day. What is the probability that both will visit the shop on (i) the same day? (ii) consecutive days?
The possible days are {Tue, Wed, Thu, Fri, Sat}. Total days = 5.
The total number of possible outcomes for both customers is 5 × 5 = 25.
(i) the same day:
Favourable outcomes are (Tue, Tue), (Wed, Wed), (Thu, Thu), (Fri, Fri), (Sat, Sat). Total = 5.
P(same day) = 5/25 = 1/5.
(ii) consecutive days:
Favourable outcomes are (Tue, Wed), (Wed, Thu), (Thu, Fri), (Fri, Sat) and also (Wed, Tue), (Thu, Wed), (Fri, Thu), (Sat, Fri). Total = 8.
P(consecutive days) = 8/25.
Answer: P(same day) = 1/5, P(consecutive days) = 8/25.
