PROBABILITY - Q&A
EXERCISE 25
1. Multiple Choice Type:
Choose the correct answer from the options given below.
(i) A coin is tossed three times, the number of possible outcomes are:
(a) 8
(b) 6
(c) 10
(d) 4
Answer: (a) 8
When a coin is tossed n times, the number of outcomes is 2n.
For 3 times, outcomes = 23 = 2 × 2 × 2 = 8.
(ii) If P(A) denotes the probability of getting an event A. Then P(not getting A) is:
(a) 1 - P(A)
(b) P(A) - 1
(c) P(A)
(d) 1 + P(A)
Answer: (a) 1 - P(A)
The sum of the probability of an event happening and not happening is 1.
P(not A) = 1 - P(A).
(iii) A coin is tossed once. The probability of getting a tail is:
(a) 1
(b) 2
(c) 1/2
(d) 0
Answer: (c) 1/2
Total outcomes = {Head, Tail} = 2.
Favourable outcome (Tail) = 1.
Probability = 1/2.
(iv) A coin is tossed two times. The probability of getting atleast one tail is:
(a) 1/4
(b) 1/3
(c) 1/2
(d) 3/4
Answer: (d) 3/4
Total outcomes = {HH, HT, TH, TT} = 4.
Favourable outcomes (at least one tail) = {HT, TH, TT} = 3.
Probability = 3/4.
(v) A card is drawn from a well shuffled pack of 52 playing cards. The probability of getting a face card is:
(a) 4/13
(b) 2/13
(c) 3/13
(d) 1/13
Answer: (c) 3/13
Total cards = 52.
Face cards (Jack, Queen, King) in 4 suits = 3 × 4 = 12.
Probability = 12/52 = 3/13.
2. A coin is tossed twice. Find the probability of getting:
(i) exactly one head
(ii) exactly one tail
(iii) two tails
(iv) two heads
Solution:
Total outcomes when tossing a coin twice = {HH, HT, TH, TT} = 4.
(i) Exactly one head: Favourable outcomes = {HT, TH} = 2.
Probability = 2/4 = 1/2.
(ii) Exactly one tail: Favourable outcomes = {HT, TH} = 2.
Probability = 2/4 = 1/2.
(iii) Two tails: Favourable outcomes = {TT} = 1.
Probability = 1/4.
(iv) Two heads: Favourable outcomes = {HH} = 1.
Probability = 1/4.
3. A letter is chosen from the word 'PENCIL', what is the probability that the letter chosen is a consonant?
Solution:
Total letters in 'PENCIL' = 6 (P, E, N, C, I, L).
Consonants are P, N, C, L. Number of consonants = 4.
Probability = 4/6 = 2/3.
4. A bag contains a black ball, a red ball and a green ball, all the balls are identical in shape and size. A ball is drawn from the bag without looking into it. What is the probability that the ball drawn is:
(i) a red ball
(ii) not a red ball
(iii) a white ball.
Solution:
Total balls = 1 (Black) + 1 (Red) + 1 (Green) = 3.
(i) Red ball: Favourable = 1. Probability = 1/3.
(ii) Not a red ball: Favourable (Black or Green) = 2. Probability = 2/3.
(iii) White ball: There are no white balls. Favourable = 0. Probability = 0/3 = 0.
5. In a single throw of a die, find the probability of getting a number
(i) greater than 2
(ii) less than or equal to 2
(iii) not greater than 2.
Solution:
Total outcomes on a die = {1, 2, 3, 4, 5, 6} = 6.
(i) Greater than 2: {3, 4, 5, 6} = 4 outcomes.
Probability = 4/6 = 2/3.
(ii) Less than or equal to 2: {1, 2} = 2 outcomes.
Probability = 2/6 = 1/3.
(iii) Not greater than 2: This is the same as "less than or equal to 2". Outcomes = {1, 2}.
Probability = 2/6 = 1/3.
6. A bag contains 3 white, 5 black and 2 red balls, all of the same size. A ball is drawn from the bag without looking into it, find the probability that the ball drawn is:
(i) a black ball
(ii) a red ball
(iii) a white ball
(iv) not a red ball
(v) not a black ball
Solution:
Total balls = 3 (White) + 5 (Black) + 2 (Red) = 10.
(i) Black ball: Favourable = 5. Probability = 5/10 = 1/2.
(ii) Red ball: Favourable = 2. Probability = 2/10 = 1/5.
(iii) White ball: Favourable = 3. Probability = 3/10.
(iv) Not a red ball: Favourable (White + Black) = 3 + 5 = 8. Probability = 8/10 = 4/5.
(v) Not a black ball: Favourable (White + Red) = 3 + 2 = 5. Probability = 5/10 = 1/2.
7. In a single throw of a die, find the probability that the number:
(i) will be an even number
(ii) will be an odd number
(iii) will not be an even number.
Solution:
Total outcomes = 6.
(i) Even number: {2, 4, 6} = 3 outcomes. Probability = 3/6 = 1/2.
(ii) Odd number: {1, 3, 5} = 3 outcomes. Probability = 3/6 = 1/2.
(iii) Not an even number: This means an odd number. Probability = 1/2.
8. In a single throw of a die, find the probability of getting:
(i) 8
(ii) a number greater than 8
(iii) a number less than 8
Solution:
Total outcomes = 6.
(i) 8: Impossible event (die has only 1-6). Probability = 0.
(ii) A number greater than 8: Impossible event. Probability = 0.
(iii) A number less than 8: Sure event (all numbers 1-6 are less than 8). Probability = 1.
9. Which of the following cannot be the probability of an event?
(i) 2/7
(ii) 3 - 8 (Note: This option appears as a typo in the book, likely meant to be a fraction or negative number)
(iii) 127%
(iv) -0.8
Answer:
The probability of an event must be between 0 and 1 (inclusive).
(iii) 127% = 1.27, which is greater than 1, so it cannot be a probability.
(iv) -0.8 is negative, so it cannot be a probability.
Thus, 127% and -0.8 cannot be probabilities.
10. A bag contains six identical black balls. A boy withdraws one ball from the bag without looking into it. What is the probability that he takes out:
(i) a white ball?
(ii) a black ball?
Solution:
Total balls = 6 (all black).
(i) White ball: There are 0 white balls. Probability = 0/6 = 0 (Impossible event).
(ii) Black ball: There are 6 black balls. Probability = 6/6 = 1 (Sure event).
11. Three identical coins are tossed together. What is the probability of obtaining:
(i) all heads?
(ii) exactly two heads?
(iii) exactly one head?
(iv) no head?
Solution:
Total outcomes = 23 = 8. Sample Space = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}.
(i) All heads: {HHH} = 1 outcome. Probability = 1/8.
(ii) Exactly two heads: {HHT, HTH, THH} = 3 outcomes. Probability = 3/8.
(iii) Exactly one head: {HTT, THT, TTH} = 3 outcomes. Probability = 3/8.
(iv) No head: {TTT} = 1 outcome. Probability = 1/8.
12. A book contains 92 pages. A page is chosen at random. What is the probability that the sum of the digits in the page number is 9?
Solution:
Total pages = 92. Total outcomes = 92.
Page numbers with digit sum 9: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90.
Total favourable outcomes = 10.
Probability = 10/92 = 5/46.
13. Two coins are tossed together. What is the probability of getting:
(i) at least one head?
(ii) both heads or both tails?
Solution:
Total outcomes = 4 {HH, HT, TH, TT}.
(i) At least one head: {HH, HT, TH} = 3 outcomes. Probability = 3/4.
(ii) Both heads or both tails: {HH, TT} = 2 outcomes. Probability = 2/4 = 1/2.
14. From 10 identical cards, numbered 1, 2, 3, ..... 10, one card is drawn at random. Find the probability that the number on the card drawn is a multiple of:
(i) 2
(ii) 3
(iii) 2 and 3
(iv) 2 or 3
Solution:
Total outcomes = 10.
(i) Multiple of 2: {2, 4, 6, 8, 10} = 5 outcomes. Probability = 5/10 = 1/2.
(ii) Multiple of 3: {3, 6, 9} = 3 outcomes. Probability = 3/10.
(iii) Multiple of 2 and 3 (i.e., multiple of 6): {6} = 1 outcome. Probability = 1/10.
(iv) Multiple of 2 or 3: {2, 3, 4, 6, 8, 9, 10} = 7 outcomes. Probability = 7/10.
15. Two dice are thrown at the same time. Find the probability that the sum of the two numbers appearing on the top of the dice is:
(i) 0
(ii) 12
(iii) less than 12
(iv) less than or equal to 12
Solution:
Total outcomes when throwing two dice = 6 × 6 = 36.
(i) Sum 0: Minimum sum is 1+1=2. Impossible event. Probability = 0.
(ii) Sum 12: Only {6, 6}. 1 outcome. Probability = 1/36.
(iii) Less than 12: All outcomes except {6, 6}. 35 outcomes. Probability = 35/36.
(iv) Less than or equal to 12: All possible sums are ≤ 12. Sure event. Probability = 36/36 = 1.
16. A die is thrown once. Find the probability of getting:
(i) a prime number
(ii) a number greater than 3
(iii) a number other than 3 and 5
(iv) a number less than 6
(v) a number greater than 6.
Solution:
Total outcomes = 6.
(i) Prime number: {2, 3, 5} = 3 outcomes. Probability = 3/6 = 1/2.
(ii) Number greater than 3: {4, 5, 6} = 3 outcomes. Probability = 3/6 = 1/2.
(iii) Other than 3 and 5: {1, 2, 4, 6} = 4 outcomes. Probability = 4/6 = 2/3.
(iv) Number less than 6: {1, 2, 3, 4, 5} = 5 outcomes. Probability = 5/6.
(v) Number greater than 6: Impossible event. Probability = 0.
17. Two coins are tossed together. Find the probability of getting:
(i) exactly one tail
(ii) at least one head
(iii) no head
(iv) at most one head
Solution:
Total outcomes = {HH, HT, TH, TT} = 4.
(i) Exactly one tail: {HT, TH} = 2 outcomes. Probability = 2/4 = 1/2.
(ii) At least one head: {HH, HT, TH} = 3 outcomes. Probability = 3/4.
(iii) No head: {TT} = 1 outcome. Probability = 1/4.
(iv) At most one head (0 or 1 head): {TT, HT, TH} = 3 outcomes. Probability = 3/4.
18. Two dice are thrown simulteneously, write all possible outcomes. Find:
(i) probability of getting same number on both the dice.
(ii) probability of getting a sum 7 on the uppermost faces of both the dice.
Solution:
Total outcomes = 36:
(1,1), (1,2), (1,3), (1,4), (1,5), (1,6)
(2,1), (2,2), (2,3), (2,4), (2,5), (2,6)
(3,1), (3,2), (3,3), (3,4), (3,5), (3,6)
(4,1), (4,2), (4,3), (4,4), (4,5), (4,6)
(5,1), (5,2), (5,3), (5,4), (5,5), (5,6)
(6,1), (6,2), (6,3), (6,4), (6,5), (6,6)
(i) Same number: {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)} = 6 outcomes.
Probability = 6/36 = 1/6.
(ii) Sum 7: {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)} = 6 outcomes.
Probability = 6/36 = 1/6.
Test yourself
1. Multiple Choice Type: Choose the correct answer from the options given below.
(i) Two dice are rolled simultaneously. The probability of getting the sum equal to 5 is:
(a) 1/6
(b) 4/9
(c) 1/9
(d) 5/6
Answer: (c) 1/9
Favourable pairs for sum 5: {(1,4), (2,3), (3,2), (4,1)} = 4 outcomes.
Probability = 4/36 = 1/9.
(ii) A dice is rolled once. The probability of getting an odd number is:
(a) 1/6
(b) 0.5
(c) 1/3
(d) none of these
Answer: (b) 0.5
Odd numbers: {1, 3, 5} = 3 outcomes.
Probability = 3/6 = 1/2 = 0.5.
(iii) A card is drawn from a well shuffled pack of 52 playing cards. The probability of getting a club card is:
(a) 3/4
(b) 2/3
(c) 1/4
(d) 1/2
Answer: (c) 1/4
There are 13 clubs in a deck.
Probability = 13/52 = 1/4.
(iv) A dice is thrown once. The probability of getting a number not more than 5 is:
(a) 1/6
(b) 1/2
(c) 1/3
(d) 5/6
Answer: (d) 5/6
Numbers not more than 5: {1, 2, 3, 4, 5} = 5 outcomes.
Probability = 5/6.
(v) A dice is rolled once. The probability of getting a prime number is:
(a) 1/2
(b) 1/3
(c) 1/4
(d) 2/3
Answer: (a) 1/2
Prime numbers: {2, 3, 5} = 3 outcomes.
Probability = 3/6 = 1/2.
(vi) Statement 1: Picking a red ball from a bag containing red balls is not a random experiment.
Statement 2: Random experiment is completely defined when we know all possible outcomes of the experiment but do not know which outcome will occur.
Which of the following options is correct?
(a) Both the statements are true.
(b) Both the statements are false.
(c) Statement 1 is true, and statement 2 is false.
(d) Statement 1 is false, and statement 2 is true.
Answer: (a) Both the statements are true.
Statement 1 is true because the outcome is certain (red), so it's not random. Statement 2 is the correct definition of a random experiment.
The following questions are Assertion-Reason based questions. Choose your answer based on the codes given below.
(1) Both A and R are correct, and R is the correct explanation for A.
(2) Both A and R are correct, and R is not the correct explanation for A.
(3) A is true, but R is false.
(4) A is false, but R is true.
(vii) Assertion (A): A dice is rolled two times the probability of getting an odd number on each dice is 1/4
Reason (R): The favourable outcomes are (1, 1), (1, 3), (1, 5), (3, 1), (3, 3), (3, 5), (5, 1), (5, 3) and (5, 6).
(a) (1)
(b) (2)
(c) (3)
(d) (4)
Answer: (c) (3)
Assertion is True: Odd on first {1,3,5} (3/6=1/2), Odd on second {1,3,5} (1/2). Total Prob = 1/2 × 1/2 = 1/4.
Reason is False: The set of outcomes listed includes (5, 6), which has an even number. Also, it excludes (5,5) which should be there.
(viii) Assertion (A): Out of the given values: 1.5/0.5, 1-1, 101% and -0.1, when asked which of them can be the probability of an event, a student answered -0.1.
Reason (R): The probability of an event always lies between 0 and 1, both inclusive.
(a) (1)
(b) (2)
(c) (3)
(d) (4)
Answer: (d) (4)
Assertion is False (in context of correctness): The student gave an incorrect answer, as probability cannot be negative. Reason is correct.
(ix) Assertion (A): When a die is thrown, the event of getting the 1st whole number is an impossible event.
Reason (R): The probability of an event always lies between 0 and 1.
(a) (1)
(b) (2)
(c) (3)
(d) (4)
Answer: (b) (2)
Assertion is True: The 1st whole number is 0. A die has {1,2,3,4,5,6}. Getting 0 is impossible.
Reason is True: Probability is between 0 and 1.
Explanation: R does not explain *why* getting 0 is impossible (which is because 0 is not in the sample space). So, (2).
(x) Assertion (A): A bag contains red, white and blue pencils. The probability of selecting a red pencil is 2/13 and that of selecting a blue pencil is 4/13. Then the probability of selecting a white pencil will be 7/13
Reason (R): The probability of all possible outcomes of an experiment must add up to 1.
(a) (1)
(b) (2)
(c) (3)
(d) (4)
Answer: (a) (1)
Assertion is True: P(W) = 1 - (P(R) + P(B)) = 1 - (2/13 + 4/13) = 1 - 6/13 = 7/13.
Reason is True and it correctly explains the calculation in A.
2. Two dice are rolled together. What is the probability of getting an odd number as sum?
Solution:
Total outcomes = 36.
For the sum to be odd, one die must be even and the other odd.
Outcomes: (1,2), (1,4), (1,6), (2,1), (2,3), (2,5), (3,2), (3,4), (3,6), (4,1), (4,3), (4,5), (5,2), (5,4), (5,6), (6,1), (6,3), (6,5).
Total favourable = 18.
Probability = 18/36 = 1/2.
3. Two dice are rolled together. What is the probability of getting a total of atleast 11?
Solution:
Total outcomes = 36.
Total of at least 11 means sum is 11 or 12.
Favourable outcomes: {(5, 6), (6, 5), (6, 6)}.
Number of outcomes = 3.
Probability = 3/36 = 1/12.
4. A card is drawn from a well shuffled deck of 52 cards. Find the probability of getting a black queen.
Solution:
Total cards = 52.
Black Queens: Queen of Spades and Queen of Clubs. Total = 2.
Probability = 2/52 = 1/26.
5. Find the probability that a leap year will have 53 Tuesdays.
Solution:
A leap year has 366 days.
366 days = 52 weeks + 2 extra days.
The 2 extra days can be: (Mon, Tue), (Tue, Wed), (Wed, Thu), (Thu, Fri), (Fri, Sat), (Sat, Sun), (Sun, Mon).
Total possibilities = 7.
Favourable for Tuesday: (Mon, Tue) and (Tue, Wed) = 2.
Probability = 2/7.
6. Numbers 1 to 10 written on ten separate identical slips (one number on one slip) are kept in box and mixed well. One slip is chosen at random from the box without looking into it. What is the probability of:
(i) getting a number less than 6?
(ii) getting a single digit number?
Solution:
Total outcomes = 10.
(i) Number less than 6: {1, 2, 3, 4, 5} = 5 outcomes.
Probability = 5/10 = 1/2.
(ii) Single digit number: {1, 2, 3, 4, 5, 6, 7, 8, 9} = 9 outcomes.
Probability = 9/10.
7. Find the probability of drawing a square number from a pack of 100 cards numbered from 1 to 100.
Solution:
Total cards = 100.
Perfect squares between 1 and 100: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.
Number of favourable outcomes = 10.
Probability = 10/100 = 1/10.
8. A dice is tossed once. What is the probability of the number 7, coming up?
Solution:
A standard die has numbers 1, 2, 3, 4, 5, 6.
Getting 7 is an impossible event.
Probability = 0.
9. A spinning wheel is divided into five equal sectors; out of which three are painted green, one is painted blue and the remaining one is painted red. What is the probability of getting a non-blue sector?
Solution:
Total sectors = 5.
Blue sectors = 1.
Non-blue sectors = Total - Blue = 5 - 1 = 4 (3 Green + 1 Red).
Probability = 4/5.
10. A box contains 21 cards numbered 1, 2, 3, 4, ....., 21 and thoroughly mixed. A card is drawn at random from this box. What is the probability that the number on the card is divisible by 2 or 3?
Solution:
Total outcomes = 21.
Divisible by 2: {2, 4, 6, 8, 10, 12, 14, 16, 18, 20} (10 numbers).
Divisible by 3: {3, 6, 9, 12, 15, 18, 21} (7 numbers).
Numbers divisible by both (multiples of 6): {6, 12, 18} (3 numbers).
Favourable outcomes = (Count of 2s) + (Count of 3s) - (Count of both)
= 10 + 7 - 3 = 14.
Probability = 14/21 = 2/3.