In Class 10 Mathematics, Probability is a crucial chapter that introduces students to the fundamentals of the chance and likelihood. It forms the basis for the understanding statistical data and making predictions based on the available information. Chapter 13 of RD Sharma's textbook delves into the various problems to the help students grasp the concept of the probability in the practical scenarios.
Probability
Probability is a branch of mathematics that deals with the likelihood or chance of different outcomes. It quantifies the chance of the specific event occurring and ranges from the 0 to 1 where 0 means the event will not happen and 1 means it is certain to happen. The basic formula for the calculating probability is:
P(\text{Probability}) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}}
This simple yet powerful formula is the foundation for solving probability problems and understanding the various random events and their outcomes.
Question 1. The probability that it will rain tomorrow is 0.85. What is the probability that it will not rain tomorrow?
Solution:
Let E be the event of raining tomorrow
P(E) = 0.85 (given)
Sum of the probability of occurrence of an event and the probability of non-occurrence of an event is 1.
P(E) + P(not E) = 1
P(not E) = 1-0.85
Therefore, the probability that it will not rain tomorrow = 0.15
Question 2. A die is thrown. Find the probability of getting:
(i) a prime number
Solution:
Total outcomes when a die is rolled are 1, 2, 3, 4, 5 and 6
Prime numbers are 2, 3, and 5
Probability = Number of favorable outcomes/ Total number of outcomes
Probability of getting a prime number = 3/6 = 1/2
(ii) 2 or 4
Solution:
Favorable outcomes = 2.
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, Probability of getting 2 or 4 = 2/6 = 1/3
(iii) a multiple of 2 or 3
Solution:
Multiple of 2 or 3 are 2, 3, 4 and 6.
Favorable outcomes = 4
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, probability of getting multiple of 2 or 3 = 4/6 = 2/3
(iv) an even prime number
Solution:
Even prime number = 2
Favorable outcomes = 1.
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting an even prime number = 1/6
(v) a number greater than 5
Solution:
Number greater than 5 = 6
Favorable outcomes = 1.
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting a number greater than 5 = 1/6
(vi) a number lying between 2 and 6
Solution:
Numbers lying between 2 and 6 are 3, 4 and 5
Favorable outcomes = 3.
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting a number lying between 2 and 6 = 3/6 =1/2
Question 3. Three coins are tossed together. Find the probability of getting:
(i) exactly two heads
Solution:
When three coins are tossed then the outcomes are TTT, THT, TTH, THH. HTT, HHT, HTH, HHH.
Total number of outcomes = 8.
Favorable outcome are THH, HHT, HTH
Number of favorable outcomes = 3.
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting exactly two heads = 3/8
(ii) at least two heads
Solution:
Favorable outcomes are HHT, HTH, HHH, THH
Number of favorable outcomes = 4.
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting at least two heads = 4/8 = 1/2
(iii) at least one head and one tail
Solution:
Favorable outcomes are THT, TTH, THH, HTT, HHT, and HTH.
Number of favorable outcomes = 6
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting at least one head and one tail = 6/8 = 3/4
(iv) no tails
Solution:
Favorable outcomes are HHH.
Number of favorable outcomes = 1.
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting no tails is 1/8.
Question 4. A and B throw a pair of dice. If A throws 9, find B’s chance of throwing a higher number
Solution:
Total outcomes are (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),
Total outcomes = 36
Favorable outcomes are (5,5), (5,6), (6,4), (4,6), (6,5) and (6,6).
Number of favorable outcomes = 6
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting the total of numbers on the dice greater than 9 = 6/36 = 1/6
Question 5. Two unbiased dice are thrown. Find the probability that the total of the numbers on the dice is greater than 10.
Solution:
Total outcomes are (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),
Total outcomes = 36
Favorable outcomes are (5, 6), (6, 5) and (6, 6).
Number of favorable events = 3.
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting the total of numbers on the dice greater than 10 = 3/36 = 1/12
Question 6. A card is drawn at random from a pack of 52 cards. Find the probability that the card drawn is:
(i) a black king
Solution:
Total number of cards = 52
Number of black king = 2
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting a black king = 2/52 = 1/26
(ii) either a black card or a king
Solution:
Total number of black cards are 26
Total number of kings which are not black = 2
Total number of black cards or king = 26+2 = 28
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting a black cards or a king = 28/52 = 7/13
(iii) black and a king
Solution:
Total number of cards which are black and a king cards = 2
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting a black cards and a king = 2/52 = 1/26
(iv) a jack, queen or a king
Solution:
A jack, queen or a king are 3 from each 4 suits.
Total number of a jack, queen and king = 12.
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting a jack, queen or a king is 12/52 = 3/13
(v) neither a heart nor a king
Solution:
Total number of cards that are a heart and a king = 13 + 3 = 16
Total number of cards that are neither a heart nor a king = 52 – 16 = 36
We know that, Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting cards neither a heart nor a king = 36/52 = 9/13
(vi) spade or an ace
Solution:
Total number of spade cards = 13
Ace other than spade = 3
Total number of card which are spade or ace = 13 + 3 = 16
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting cards that is spade or an ace = 16/52 = 4/13
(vii) neither an ace nor a king
Solution:
Total number of cards that are an ace or a king = 4 + 4 = 8
Total cards that are neither an ace nor a king = 52 – 8 = 44
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting cards which are neither an ace nor a king = 44/52 = 11/13
(viii) neither a red card nor a queen
Solution:
Total red cards = 26.
Queens which are not red = 2
Total number of red cards or queen = 26 + 2 = 28
Total number of cards that are neither a red nor a queen= 52 -28 = 24
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting neither a red card nor a queen = 24/52 = 6/13
(ix) other than ace
Solution:
Total number of ace = 4
Cards other than ace = 52-4 = 48
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of cards other than ace = 48/52 = 12/13
(x) a ten
Solution:
Total number of tens = 4.
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting a ten = 4/52 = 1/13
(xi) a spade
Solution:
Total number of spade = 13
Probability = Number of favorable outcomes/ Total number of outcomes
Thus, the probability of getting a spade = 13/52 = 1/4
(xii) a black card
Solution:
Total number of black cards = 26
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting black cards = 26/52 = 1/2
(xiii) a seven of clubs
Solution:
Total number of 7 of club = 1
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting a 7 of club = 1/52
(xiv) jack
Solution:
Total number of jacks = 4
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting a jack = 4/52 = 1/13
(xv) the ace of spades
Solution:
Total number of ace of spade = 1
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting an ace of spade = 1/52
(xvi) a queen
Solution:
Total number of queens = 4
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting a queen = 4/52 = 1/13
(xvii) a heart
Solution:
Total number of heart cards = 13
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting a heart card = 13/52 = 1/4
(xviii) a red card
Solution:
Total number of red cards = 26
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting a red card = 26/52 = 1/2
Question 7. In a lottery of 50 tickets numbered 1 to 50, one ticket is drawn. Find the probability that the drawn ticket bears a prime number.
Solution:
Total number of tickets = 50.
Tickets which are numbered as prime number are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
Favorable outcomes = 15.
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting a prime number on the ticket = 15/50 = 3/10
Question 8. An urn contains 10 red and 8 white balls. One ball is drawn at random. Find the probability that the ball drawn is white.
Solution:
Total number of balls = 10 + 8 = 18
Total number of white balls = 8
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of drawing a white ball from the urn is 8/18 = 4/9
Question 9. A bag contains 3 red balls, 5 black balls and 4 white balls. A ball is drawn at random from the bag. What is the probability that the ball drawn is:
(i) white?
Solution:
Total number of balls = 3 + 5 + 4 =12
Total white balls = 4
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting a white ball = 4/12 = 1/3
(ii) red?
Solution:
Total red balls = 3
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting a red ball = 3/12 = 1/4
(iii) black?
Solution:
Total black balls is 5
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting a black ball = 5/12
(iv) not red
Solution:
Number of balls which are not red are 4 white balls and 5 black balls = 4 + 5 = 9
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting no red ball = 9/12 = 3/4
Question 10. What is the probability that a number selected from the numbers 1, 2, 3, …, 15 is a multiple of 4?
Solution:
Total numbers = 15
Numbers that are multiple of 4 are 4, 8 and 12.
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of selecting a number which a multiple of 4 = 3/15 = 1/5
Question 11. A bag contains 6 red, 8 black and 4 white balls. A ball is drawn at random. What is the probability that the ball drawn is not black?
Solution:
Total number of balls = 6 + 8 + 4 = 18
Total black balls = 8
Number of balls which are not black is = 18 – 8 = 10
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of drawing a ball which is not black = 10/18 = 5/9
Question 12. A bag contains 5 white balls and 7 red balls. One ball is drawn at random. What is the probability that ball drawn is white?
Solution:
Total balls = 7 + 5 = 12
Total number of white balls = 5
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting a white ball = 5/12
Question 13. Tickets numbered from 1 to 20 are mixed up and a ticket is drawn at random. What is the probability that the ticket drawn has a number which is a multiple of 3 or 7?
Solution:
Total number of cards = 20.
Multiple of 3 or 7 are 3, 6, 7, 9, 12, 14, 15 and 18.
Favorable outcomes = 8.
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of drawing a card that is a multiple of 3 or 7 = 8/20 = 2/5
Question 14. In a lottery, there are 10 prizes and 25 blanks. What is the probability of getting a prize?
Solution:
Total number of tickets = 10 + 25 = 35
Number of prize carrying tickets = 10
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of winning a prize = 10/35 = 2/7
Question 15. If the probability of winning a game is 0.3, what is the probability of losing it?
Solution:
Sum of probability of occurrence of an event and probability of non-occurrence of an event is 1.
Therefore, P(E) + P(not E) = 1
P(not E) = 1- 0.3
P(not E) = 0.7
Therefore, the probability of losing the game = 0.7
Question 16. A bag contains 5 black, 7 red and 3 white balls. A ball is drawn from the bag at random. Find the probability that the ball drawn is:
(i) red
Solution:
Total balls = 7 + 5 + 3 = 15
Number red balls = 7
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of drawing a red ball = 7/15
(ii) black or white
Solution:
Total number of black or white balls = 5 + 3 = 8
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of drawing white or black ball = 8/15
(iii) not black
Solution:
Total number of black balls = 5
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of drawing black ball P(E) = 5/15 = 1/3
Therefore, probability of not black = 1 - P(E)
= 1 - 1/3
= 2/3
Question 17. A bag contains 4 red, 5 black and 6 white balls. A ball is drawn from the bag at random. Find the probability that the ball drawn is:
(i) White
Solution:
Total number of balls = 4 + 5 + 6 = 15
Total white balls = 6
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of drawing white a ball = 6/15 = 2/5
(ii) Red
Solution:
Total number of red balls = 4
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of drawing red a ball = 4/15
(iii) Not black
Solution:
Total number of black balls = 5
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of drawing black ball P(E) = 5/15 = 1/3
Therefore, probability of not black = 1 - P(E)
= 1 - 1/3
= 2/3
Therefore, the probability of drawing a ball that is not black is 2/3
(iv) Red or White
Solution:
Total number of red or white balls= 4 + 6 = 10
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of drawing a white or red ball = 10/15 = 2/3
Question 18. One card is drawn from a well-shuffled deck of 52 cards. Find the probability of getting:
(i) A king of red suit
Solution:
Total number of cards = 52
Total cards which are king of red suit = 2
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting cards which is a king of red suit = 2/52 = 1/26
(ii) A face card
Solution:
Total number of face cards = 12
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting a face card = 12/52 = 3/13
(iii) A red face card
Solution:
Total number of red face cards = 6
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting a red face card = 6/52 = 3/26
(iv) A queen of black suit
Solution:
Total number of queen of black suit cards = 2
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting cards which is a queen of black suit = 2/52 = 1/26
(v) A jack of hearts
Solution:
Total number of jack of hearts = 1
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting a card which is a jack of hearts = 1/52
(vi) A spade
Solution:
Total number of spade cards = 13
Probability = Number of favorable outcomes/ Total number of outcomes
Therefore, the probability of getting a spade card = 13/52 = 1/4
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