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CBSE Class 10 Maths Chapter 14 Probability Model Questions - 2 Marks - Part 2

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SECTION B — Very Short Answer Type Questions [2 Marks Each]

• Q26. A card is drawn from a well-shuffled deck of 52 cards. Find the probability of getting a spade or an ace.
  Answer: $\frac{4}{13}$
  
  Justification & Calculation:
  Total cards = $52$.
  
  Number of spades = $13$. Number of aces = $4$. The ace of spades is counted in both groups.
  Favorable outcomes = $13 + 4 - 1 = 16$.
  $$\text{P}(E) = \frac{16}{52} = \frac{4}{13}$$
• Q27. Five cards—the ten, jack, queen, king, and ace of diamonds—are well-shuffled with their face downwards. One card is picked up at random. What is the probability that the card is the queen?
  Answer: $\frac{1}{5}$
  
  Justification & Calculation:
  Total number of cards = $5$.
  
  Number of queens = $1$.
  $$\text{P}(E) = \frac{1}{5}$$
• Q28. If the queen of diamonds is removed from the well-shuffled pack of 52 cards, find the probability of drawing a diamond.
  Answer: $\frac{3}{13}$
  
  Justification & Calculation:
  Total remaining cards = $52 - 1 = 51$.
  
  Total remaining diamonds = $13 - 1 = 12$.
  $$\text{P}(E) = \frac{12}{51} = \frac{4}{13}$$
• Q29. A card is drawn from a well-shuffled deck of 52 cards. Find the probability of getting a face card which is a club.
  Answer: $\frac{3}{52}$
  
  Justification & Calculation:
  Total outcomes = $52$.
  
  Face cards that are clubs (Jack, Queen, King of clubs) = $3$.
  $$\text{P}(E) = \frac{3}{52}$$
• Q30. From a pack of 52 cards, all face cards are removed. Find the probability of drawing a red card from the remaining deck.
  Answer: $\frac{1}{2}$
  
  Justification & Calculation:
  Total face cards removed = $12$. Remaining cards = $52 - 12 = 40$.
  
  Red face cards removed = $6$. Remaining red cards = $26 - 6 = 20$.
  $$\text{P}(E) = \frac{20}{40} = \frac{1}{2}$$

• Q31. A card is drawn from a well-shuffled deck of 52 cards. Find the probability of getting a non-face card.
  Answer: $\frac{10}{13}$
  
  Justification & Calculation:
  Total number of cards = $52$.
  
  Total number of face cards = $12$.
  Number of non-face cards = $52 - 12 = 40$.
  $$\text{P}(E) = \frac{40}{52} = \frac{10}{13}$$
• Q32. The king, queen, and jack of clubs are removed from a deck of 52 playing cards and the remaining cards are well-shuffled. A card is drawn at random. Find the probability of getting a club.
  Answer: $\frac{10}{49}$
  
  Justification & Calculation:
  Total cards removed = $3$ (King, Queen, Jack of clubs).
  Remaining total cards = $52 - 3 = 49$.
  
  Total clubs originally = $13$. Remaining clubs = $13 - 3 = 10$.
  $$\text{P}(E) = \frac{10}{49}$$
• Q33. A card is drawn from a well-shuffled deck of 52 cards. Find the probability of getting a number card divisible by 3.
  Answer: $\frac{3}{13}$
  
  Justification & Calculation:
  Total cards = $52$.
  
  Number cards divisible by $3$ in each suit ($2$ to $10$) are $3, 6, 9$ (3 cards per suit).
  Total favorable outcomes = $3 \times 4 = 12$.
  $$\text{P}(E) = \frac{12}{52} = \frac{3}{13}$$
• Q34. From a deck of 52 cards, red jacks and red queens are removed. Find the probability of drawing a black card.
  Answer: $\frac{13}{24}$
  
  Justification & Calculation:
  Total cards removed = $2$ red jacks + $2$ red queens = $4$ cards.
  Remaining total cards = $52 - 4 = 48$.
  
  Black cards are unaffected and remain $26$.
  $$\text{P}(E) = \frac{26}{48} = \frac{13}{24}$$
• Q35. A card is drawn at random from a pack of 52 cards. Find the probability that the card drawn is neither an ace nor a king.
  Answer: $\frac{11}{13}$
  
  Justification & Calculation:
  Total cards = $52$.
  
  Total aces and kings = $4 + 4 = 8$.
  Cards that are neither = $52 - 8 = 44$.
  $$\text{P}(E) = \frac{44}{52} = \frac{11}{13}$$
• Q36. A card is drawn from a well-shuffled deck of 52 cards. Find the probability of getting a card of black color and a face card.
  Answer: $\frac{3}{26}$
  
  Justification & Calculation:
  Total cards = $52$.
  
  Black face cards = $3$ in spades + $3$ in clubs = $6$ face cards.
  $$\text{P}(E) = \frac{6}{52} = \frac{3}{26}$$
• Q37. A bag contains 5 red balls and some blue balls. If the probability of drawing a blue ball is double that of a red ball, find the number of blue balls in the bag.
  Answer: $10$
  
  Justification & Calculation:
  Let the number of blue balls be $x$. Total balls = $5 + x$.
  Probability of red ball = $\frac{5}{5 + x}$.
  Probability of blue ball = $\frac{x}{5 + x}$.
  
  Given: $\frac{x}{5 + x} = 2 \times \frac{5}{5 + x} \implies x = 10$.
  Number of blue balls = $10$.
• Q38. A box contains 90 discs which are numbered from 1 to 90. If one disc is drawn at random from the box, find the probability that it bears a two-digit number.
  Answer: $\frac{9}{10}$
  
  Justification & Calculation:
  Total number of outcomes = $90$.
  
  Two-digit numbers range from $10$ to $90$, which are $90 - 10 + 1 = 81$ numbers.
  $$\text{P}(E) = \frac{81}{90} = \frac{9}{10}$$
• Q39. A box contains 12 balls out of which $x$ are black. If one ball is drawn at random, what is the probability that it is black? If 6 more black balls are put in the box, the probability of drawing a black ball is now double of what it was before. Find $x$.
  Answer: $3$
  
  Justification & Calculation:
  Initial probability of getting a black ball = $\frac{x}{12}$.
  When $6$ more black balls are added, total black balls = $x + 6$, and total balls = $12 + 6 = 18$.
  New probability = $\frac{x + 6}{18}$.
  
  Given: $\frac{x + 6}{18} = 2 \times \frac{x}{12} \implies \frac{x + 6}{18} = \frac{x}{6} \implies 3(x + 6) = 18x \implies 6x = 18 \implies x = 3$.
• Q40. A game of chance consists of spinning an arrow which comes to rest pointing at one of the numbers $1, 2, 3, 4, 5, 6, 7, 8$, and these are equally likely outcomes. Find the probability that the arrow will point at a factor of 8.
  Answer: $\frac{1}{2}$
  
  Justification & Calculation:
  Total outcomes = $8$ ($\{1, 2, 3, 4, 5, 6, 7, 8\}$).
  
  Factors of $8$ are $\{1, 2, 4, 8\}$ (4 outcomes).
  $$\text{P}(E) = \frac{4}{8} = \frac{1}{2}$$
• Q41. A box contains 100 cards numbered from 1 to 100. If a card is drawn at random from the box, find the probability that it bears a number which is a perfect square.
  Answer: $\frac{1}{10}$
  
  Justification & Calculation:
  Total outcomes = $100$.
  
  Perfect squares from $1$ to $100$ are $\{1, 4, 9, 16, 25, 36, 49, 64, 81, 100\}$ ($10$ numbers).
  $$\text{P}(E) = \frac{10}{100} = \frac{1}{10}$$
• Q42. Savita and Hamida are friends. What is the probability that both will have different birthdays (ignoring a leap year)?
  Answer: $\frac{364}{365}$
  
  Justification & Calculation:
  Total days in a year = $365$.
  
  Probability of same birthday = $\frac{1}{365}$.
  Probability of different birthdays = $1 - \frac{1}{365} = \frac{364}{365}$.
• Q43. A piggy bank contains one 50p coin, fifty ₹1 coins, twenty ₹2 coins, and ten ₹5 coins. If it is equally likely that one of the coins will fall out when the bank is turned upside down, what is the probability that the coin will be a 50p coin?
  Answer: $\frac{1}{81}$
  
  Justification & Calculation:
  Total coins = $1 + 50 + 20 + 10 = 81$.
  
  Number of 50p coins = $1$.
  $$\text{P}(E) = \frac{1}{81}$$
• Q44. A lot consists of 144 ballpens of which 20 are defective and the others are good. Nuri will buy a pen if it is good, but will not buy if it is defective. The shopkeeper draws one pen at random and gives it to her. What is the probability that she will buy it?
  Answer: $\frac{31}{36}$
  
  Justification & Calculation:
  Total pens = $144$.
  
  Defective pens = $20$. Good pens = $144 - 20 = 124$.
  $$\text{P}(E) = \frac{124}{144} = \frac{31}{36}$$
• Q45. A bag contains 3 red balls and 5 black balls. A ball is drawn at random from the bag. What is the probability that the ball drawn is not red?
  Answer: $\frac{5}{8}$
  
  Justification & Calculation:
  Total balls = $3 + 5 = 8$.
  
  Number of non-red (black) balls = $5$.
  $$\text{P}(E) = \frac{5}{8}$$
• Q46. A box contains 5 red marbles, 8 white marbles, and 4 green marbles. One marble is taken out of the box at random. What is the probability that the marble taken out is not white?
  Answer: $\frac{9}{17}$
  
  Justification & Calculation:
  Total marbles = $5 + 8 + 4 = 17$.
  
  Marbles that are not white = $5 \text{ (red)} + 4 \text{ (green)} = 9$.
  $$\text{P}(E) = \frac{9}{17}$$
• Q47. Find the probability that a number selected at random from the numbers $1, 2, 3, \dots, 35$ is a prime number.
  Answer: $\frac{11}{35}$
  
  Justification & Calculation:
  Total numbers = $35$.
  
  Prime numbers between $1$ and $35$ are $\{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31\}$ ($11$ numbers).
  $$\text{P}(E) = \frac{11}{35}$$
• Q48. A carton consists of 100 shirts of which 88 are good, 8 have minor defects, and 4 have major defects. Jimmy, a trader, will only accept the shirts which are good, but Sujatha, another trader, will only reject the shirts with major defects. One shirt is drawn at random from the carton. What is the probability that it is acceptable to Jimmy?
  Answer: $\frac{22}{25}$
  
  Justification & Calculation:
  Total shirts = $100$.
  
  Shirts acceptable to Jimmy (good shirts) = $88$.
  $$\text{P}(E) = \frac{88}{100} = \frac{22}{25}$$
• Q49. Cards marked with numbers $2$ to $101$ are placed in a box and mixed thoroughly. One card is drawn from this box. Find the probability that the number on the card is an even number.
  Answer: $\frac{1}{2}$
  
  Justification & Calculation:
  Total numbers from $2$ to $101$ = $101 - 2 + 1 = 100$.
  
  Even numbers in this range = $50$.
  $$\text{P}(E) = \frac{50}{100} = \frac{1}{2}$$
• Q50. A box contains cards numbered $1$ to $50$. A card is drawn at random from the box. Find the probability that the number on the drawn card is a prime number less than 20.
  Answer: $\frac{2}{25}$
  
  Justification & Calculation:
  Total outcomes = $50$.
  
  Prime numbers less than $20$ are $\{2, 3, 5, 7, 11, 13, 17, 19\}$ ($8$ numbers).
  $$\text{P}(E) = \frac{8}{50} = \frac{4}{25}$$

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