Probability - Solutions

Probability - 1 Mark Solutions

Important Probability Concepts:

• Probability = Number of favorable outcomes / Total number of outcomes
• 0 ≤ P(E) ≤ 1
• P(E) + P(not E) = 1
• Prime numbers: Numbers with exactly two factors (1 and itself)
• Composite numbers: Numbers with more than two factors
• Equally likely events: Events with same probability of occurrence

When a die is rolled once unbiased, what is the probability of getting a multiple of 3 out of possible outcomes? (M'15)

Step 1: Identify total outcomes

When a die is rolled, total possible outcomes = 6
Sample space = {1, 2, 3, 4, 5, 6}

Step 2: Identify favorable outcomes

Multiples of 3 from 1 to 6: {3, 6}
Number of favorable outcomes = 2

Step 3: Apply probability formula

Probability = Favorable outcomes / Total outcomes
P(multiple of 3) = 2/6 = 1/3

∴ The probability of getting a multiple of 3 is 1/3

The probability of an event is always in between 0 and 1 why? (J'15)

Step 1: Understand probability definition

Probability = Number of favorable outcomes / Total number of outcomes

Step 2: Analyze numerator and denominator

- Number of favorable outcomes is always ≥ 0
- Number of favorable outcomes is always ≤ Total number of outcomes
- Total number of outcomes is always ≥ 1

Step 3: Establish the range

Minimum value: When favorable outcomes = 0
P(E) = 0/Total outcomes = 0

Maximum value: When favorable outcomes = Total outcomes
P(E) = Total outcomes/Total outcomes = 1

For any other case: 0 < P(E) < 1

∴ Probability always lies between 0 and 1 because favorable outcomes can't be negative and can't exceed total outcomes.

Find the probability of getting a sum of the numbers on them is 7, when two dice are rolled at a time. (M'16)

Step 1: Identify total outcomes

When two dice are rolled:
Total outcomes = 6 × 6 = 36

Step 2: Identify favorable outcomes

Pairs with sum 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1)
Number of favorable outcomes = 6

Step 3: Apply probability formula

Probability = Favorable outcomes / Total outcomes
P(sum = 7) = 6/36 = 1/6

∴ The probability of getting a sum of 7 is 1/6

Find the probability of getting a prime number, when a card drawn at random from the numbered cards from 1 to 25. (J'16)

Step 1: Identify total outcomes

Cards numbered from 1 to 25
Total outcomes = 25

Step 2: Identify favorable outcomes

Prime numbers from 1 to 25: 2, 3, 5, 7, 11, 13, 17, 19, 23
Number of favorable outcomes = 9

Step 3: Apply probability formula

Probability = Favorable outcomes / Total outcomes
P(prime) = 9/25

∴ The probability of getting a prime number is 9/25

From the first 50 natural numbers, find the probability of randomly selected number is a multiple of 3. (M'17)

Step 1: Identify total outcomes

First 50 natural numbers: 1 to 50
Total outcomes = 50

Step 2: Identify favorable outcomes

Multiples of 3 from 1 to 50: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48
Number of favorable outcomes = 16

Step 3: Apply probability formula

Probability = Favorable outcomes / Total outcomes
P(multiple of 3) = 16/50 = 8/25

∴ The probability of getting a multiple of 3 is 8/25

A dice is thrown once. Find the probability of getting a composite number. (J'17)

Step 1: Identify total outcomes

When a die is thrown, total possible outcomes = 6
Sample space = {1, 2, 3, 4, 5, 6}

Step 2: Identify favorable outcomes

Composite numbers from 1 to 6: 4, 6
(Note: 1 is neither prime nor composite)
Number of favorable outcomes = 2

Step 3: Apply probability formula

Probability = Favorable outcomes / Total outcomes
P(composite) = 2/6 = 1/3

∴ The probability of getting a composite number is 1/3

What is the probability of getting exactly 2 heads, when three coins are tossed simultaneously. (M'18)

Step 1: Identify total outcomes

When three coins are tossed:
Total outcomes = 2 × 2 × 2 = 8
Sample space = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}

Step 2: Identify favorable outcomes

Outcomes with exactly 2 heads: HHT, HTH, THH
Number of favorable outcomes = 3

Step 3: Apply probability formula

Probability = Favorable outcomes / Total outcomes
P(exactly 2 heads) = 3/8

∴ The probability of getting exactly 2 heads is 3/8

When a dice is rolled, find the probability of getting an odd prime number. (J'18)

Step 1: Identify total outcomes

When a die is rolled, total possible outcomes = 6
Sample space = {1, 2, 3, 4, 5, 6}

Step 2: Identify favorable outcomes

Odd prime numbers from 1 to 6: 3, 5
(Note: 2 is prime but not odd)
Number of favorable outcomes = 2

Step 3: Apply probability formula

Probability = Favorable outcomes / Total outcomes
P(odd prime) = 2/6 = 1/3

∴ The probability of getting an odd prime number is 1/3

From English alphabet if a letter is chosen at random, then find the probability that the letter is a consonant. (M'19)

Step 1: Identify total outcomes

Total letters in English alphabet = 26

Step 2: Identify favorable outcomes

Vowels: A, E, I, O, U (5 vowels)
Consonants = Total letters - Vowels = 26 - 5 = 21

Step 3: Apply probability formula

Probability = Favorable outcomes / Total outcomes
P(consonant) = 21/26

∴ The probability of getting a consonant is 21/26

Write two examples for equally likely events. (J'19)

Step 1: Understand equally likely events

Equally likely events are events that have the same probability of occurrence.

Step 2: Example 1

When a fair coin is tossed:
P(Head) = 1/2 and P(Tail) = 1/2
Getting a Head and getting a Tail are equally likely events.

Step 3: Example 2

When a fair die is rolled:
P(1) = P(2) = P(3) = P(4) = P(5) = P(6) = 1/6
Getting any specific number from 1 to 6 are equally likely events.

∴ Two examples of equally likely events are: (1) Getting Head or Tail when tossing a fair coin, (2) Getting any number from 1 to 6 when rolling a fair die.

If a dice rolled once, then find the probability of getting an odd number. (May 2022)

Step 1: Identify total outcomes

When a die is rolled, total possible outcomes = 6
Sample space = {1, 2, 3, 4, 5, 6}

Step 2: Identify favorable outcomes

Odd numbers from 1 to 6: 1, 3, 5
Number of favorable outcomes = 3

Step 3: Apply probability formula

Probability = Favorable outcomes / Total outcomes
P(odd) = 3/6 = 1/2

∴ The probability of getting an odd number is 1/2

If an unbiased dice is rolled once, then find the probability of getting a prime number on its top face. (Aug 22)

Step 1: Identify total outcomes

When a die is rolled, total possible outcomes = 6
Sample space = {1, 2, 3, 4, 5, 6}

Step 2: Identify favorable outcomes

Prime numbers from 1 to 6: 2, 3, 5
(Note: 1 is not a prime number)
Number of favorable outcomes = 3

Step 3: Apply probability formula

Probability = Favorable outcomes / Total outcomes
P(prime) = 3/6 = 1/2

∴ The probability of getting a prime number is 1/2

Find the probability of getting a 'vowel' if a letter is chosen randomly from the word "INNOVATION". (Apr'23)

Step 1: Identify total outcomes

Letters in "INNOVATION": I, N, N, O, V, A, T, I, O, N
Total letters = 10

Step 2: Identify favorable outcomes

Vowels in "INNOVATION": I, O, A, I, O
Number of vowels = 5

Step 3: Apply probability formula

Probability = Favorable outcomes / Total outcomes
P(vowel) = 5/10 = 1/2

∴ The probability of getting a vowel is 1/2

Probability (2 Marks) - Complete Solutions

Important Probability Concepts:

• Probability = Number of favorable outcomes / Total number of outcomes
• P(not E) = 1 - P(E)
• Composite numbers: Numbers with more than two factors
• Prime numbers: Numbers with exactly two factors (1 and itself)
• Face cards: Jack, Queen, King (12 in a deck)
• Aces: 4 in a deck

There are 12 red balls, 18 blue balls and 6 white balls in a box. When a ball is drawn at random from the box, what is the probability of not getting a red ball? (M'15)

Step 1: Find total number of balls

Red balls = 12
Blue balls = 18
White balls = 6
Total balls = 12 + 18 + 6 = 36

Step 2: Find probability of getting a red ball

P(red) = Number of red balls / Total balls = 12/36 = 1/3

Step 3: Find probability of not getting a red ball

P(not red) = 1 - P(red) = 1 - 1/3 = 2/3

∴ The probability of not getting a red ball is 2/3

When a card is drawn from a well shuffled deck of 52 cards, then find the probability of NOT getting a red faced card. (J'15)

Step 1: Understand the deck composition

Total cards = 52
Face cards: Jack, Queen, King (3 per suit)
Red suits: Hearts and Diamonds (2 suits)
Red face cards = 3 × 2 = 6

Step 2: Find probability of getting a red face card

P(red face card) = 6/52 = 3/26

Step 3: Find probability of not getting a red face card

P(not red face card) = 1 - P(red face card) = 1 - 3/26 = 23/26

∴ The probability of not getting a red face card is 23/26

There are 5 red balls, 4 green balls and 6 yellow balls in a box. If a ball is selected at random, what is the probability of not getting a yellow ball? (J'16)

Step 1: Find total number of balls

Red balls = 5
Green balls = 4
Yellow balls = 6
Total balls = 5 + 4 + 6 = 15

Step 2: Find probability of getting a yellow ball

P(yellow) = Number of yellow balls / Total balls = 6/15 = 2/5

Step 3: Find probability of not getting a yellow ball

P(not yellow) = 1 - P(yellow) = 1 - 2/5 = 3/5

∴ The probability of not getting a yellow ball is 3/5

What is the probability of a number picked from first 20 natural numbers is even composite number? (M'18)

Step 1: Identify total outcomes

First 20 natural numbers: 1 to 20
Total outcomes = 20

Step 2: Identify favorable outcomes

Even composite numbers from 1 to 20: 4, 6, 8, 10, 12, 14, 16, 18, 20
(Note: 2 is prime, not composite)
Number of favorable outcomes = 9

Step 3: Apply probability formula

P(even composite) = 9/20

∴ The probability of getting an even composite number is 9/20

A bag contains 7 red, 5 white and 6 black balls. A ball is drawn from the bag at random; find the probability that the ball drawn is not black. (J'18)

Step 1: Find total number of balls

Red balls = 7
White balls = 5
Black balls = 6
Total balls = 7 + 5 + 6 = 18

Step 2: Find probability of getting a black ball

P(black) = Number of black balls / Total balls = 6/18 = 1/3

Step 3: Find probability of not getting a black ball

P(not black) = 1 - P(black) = 1 - 1/3 = 2/3

∴ The probability that the ball drawn is not black is 2/3

A bag contains balls which are numbered from 1 to 50. A ball is drawn at random from the bag, the probability that it bears two digit number multiple of 7. (M'19)

Step 1: Identify total outcomes

Numbers from 1 to 50
Total outcomes = 50

Step 2: Identify favorable outcomes

Two-digit multiples of 7: 14, 21, 28, 35, 42, 49
Number of favorable outcomes = 6

Step 3: Apply probability formula

P(two-digit multiple of 7) = 6/50 = 3/25

∴ The probability of getting a two-digit number multiple of 7 is 3/25

A box contains 4 red balls, 5 green balls and P white balls. If the probability of randomly picked a ball from the box to be red ball is 1/3, then find the number of white balls. (J'19)

Step 1: Set up the equation

Red balls = 4
Green balls = 5
White balls = P
Total balls = 4 + 5 + P = 9 + P
P(red) = 4/(9 + P) = 1/3

Step 2: Solve for P

4/(9 + P) = 1/3
Cross multiply: 4 × 3 = 1 × (9 + P)
12 = 9 + P
P = 12 - 9 = 3

∴ The number of white balls is 3

A bag contains 5 red, 8 white, 4 green colour balls. If a ball is selected randomly from the bag then find the probability that selected ball is (i) a green ball (ii) not white ball.

Step 1: Find total number of balls

Red balls = 5
White balls = 8
Green balls = 4
Total balls = 5 + 8 + 4 = 17

(i) Probability of green ball

P(green) = Number of green balls / Total balls = 4/17

(ii) Probability of not white ball

P(white) = 8/17
P(not white) = 1 - P(white) = 1 - 8/17 = 9/17

∴ (i) P(green) = 4/17, (ii) P(not white) = 9/17

A box contains four slips numbered 1, 2, 3, 4 and another box contains five slips numbered 5, 6, 7, 8, 9. If one slip is taken randomly from each box, i) How many number pairs are possible? ii) What is the probability of both being odd? iii) What is the probability of getting the sum of the numbers 10? (Apr'23)

Step 1: Total possible pairs

First box: 4 slips
Second box: 5 slips
Total pairs = 4 × 5 = 20

(i) Number of possible pairs

Total pairs = 20

(ii) Probability of both being odd

Odd numbers in first box: 1, 3 (2 numbers)
Odd numbers in second box: 5, 7, 9 (3 numbers)
Favorable pairs = 2 × 3 = 6
P(both odd) = 6/20 = 3/10

(iii) Probability of sum being 10

Pairs with sum 10: (1,9), (2,8), (3,7), (4,6)
Number of favorable pairs = 4
P(sum = 10) = 4/20 = 1/5

∴ (i) 20 pairs, (ii) P(both odd) = 3/10, (iii) P(sum = 10) = 1/5

If one card is randomly selected from a well shuffled deck of cards, then find the probability of getting- i) a face card, (ii) a jack of hearts and (iii) an ace card. (Jun'23)

Step 1: Understand deck composition

Total cards = 52
Face cards: Jack, Queen, King (3 per suit × 4 suits = 12)
Aces: 1 per suit × 4 suits = 4

(i) Probability of face card

P(face card) = 12/52 = 3/13

(ii) Probability of jack of hearts

There is only 1 jack of hearts
P(jack of hearts) = 1/52

(iii) Probability of ace card

P(ace) = 4/52 = 1/13

∴ (i) P(face card) = 3/13, (ii) P(jack of hearts) = 1/52, (iii) P(ace) = 1/13

Probability (4 Marks) - Complete Solutions

Important Probability Concepts:

• Probability = Number of favorable outcomes / Total number of outcomes
• P(not E) = 1 - P(E)
• Prime numbers: Numbers with exactly two factors (1 and itself)
• Composite numbers: Numbers with more than two factors
• Face cards: Jack, Queen, King (12 in a deck)
• Aces: 4 in a deck

There are 100 flash cards labeled from 1 to 100 in a bag. When a card is drawn from the bag at random, what is the probability of getting…… i) A card with prime number from possible outcomes ii) A card without prime number from possible outcomes. (M'15)

Step 1: Find total outcomes

Total cards = 100

(i) Probability of prime number

Step 2: Find prime numbers between 1 and 100

Prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97
Total prime numbers = 25

Step 3: Calculate probability

P(prime) = 25/100 = 1/4

(ii) Probability of not prime number

Step 4: Calculate probability

P(not prime) = 1 - P(prime) = 1 - 1/4 = 3/4
OR
Non-prime numbers = 100 - 25 = 75
P(not prime) = 75/100 = 3/4

∴ (i) P(prime) = 1/4, (ii) P(not prime) = 3/4

A shopkeeper has 100 memory cards in a box. Among them, 15 memory cards are defective. When a person came to the shop to buy a memory card, the shopkeeper drew a memory card at random from the box. Then What is the probability that this memory card is defective? After drawing the first memory card which is defective, it is not placed back in the box. Then another memory card is drawn at random. What is the probability that this memory card is NOT defective? (J'15)

(i) Probability first card is defective

Step 1: Find total outcomes

Total cards = 100
Defective cards = 15
P(defective) = 15/100 = 3/20

(ii) Probability second card is not defective

Step 2: Update counts after first draw

First card drawn is defective
Remaining cards = 100 - 1 = 99
Remaining defective cards = 15 - 1 = 14
Non-defective cards = 99 - 14 = 85

Step 3: Calculate probability

P(second card not defective) = 85/99

∴ (i) P(defective) = 3/20, (ii) P(second card not defective) = 85/99

A bag contains 5 red balls and some blue balls. If the probability of drawing a blue ball is double that of red ball, find the number of blue balls in the bag. (M'16)

Step 1: Define variables

Let number of blue balls = x
Total balls = 5 + x

Step 2: Write probability expressions

P(red) = 5/(5 + x)
P(blue) = x/(5 + x)

Step 3: Set up equation

Given: P(blue) = 2 × P(red)
x/(5 + x) = 2 × 5/(5 + x)
x/(5 + x) = 10/(5 + x)

Step 4: Solve for x

x = 10 (since denominators are equal and non-zero)

∴ The number of blue balls is 10

Two dice are rolled at same time and the sum of the numbers appearing on them is noted. Find the probability of getting each sum, from 3 to 5 separately. (J'16)

Step 1: Find total outcomes

When two dice are rolled:
Total outcomes = 6 × 6 = 36

Probability of sum = 3

Step 2: Find favorable outcomes for sum 3

Pairs with sum 3: (1,2), (2,1)
Number of favorable outcomes = 2
P(sum = 3) = 2/36 = 1/18

Probability of sum = 4

Step 3: Find favorable outcomes for sum 4

Pairs with sum 4: (1,3), (2,2), (3,1)
Number of favorable outcomes = 3
P(sum = 4) = 3/36 = 1/12

Probability of sum = 5

Step 4: Find favorable outcomes for sum 5

Pairs with sum 5: (1,4), (2,3), (3,2), (4,1)
Number of favorable outcomes = 4
P(sum = 5) = 4/36 = 1/9

∴ P(sum = 3) = 1/18, P(sum = 4) = 1/12, P(sum = 5) = 1/9

A bag contains some square cards. A prime number between 1 and 100 has been written on each card. Find the probability of getting a card that the sum of the digits of a prime number written on it, is 8. (M'17)

Step 1: Find total prime numbers between 1 and 100

Prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97
Total prime numbers = 25

Step 2: Find primes with digit sum = 8

Check each prime:
17 → 1 + 7 = 8 ✓
53 → 5 + 3 = 8 ✓
71 → 7 + 1 = 8 ✓
No other primes between 1-100 have digit sum 8
Favorable outcomes = 3

Step 3: Calculate probability

P(digit sum = 8) = 3/25

∴ The probability of getting a card with prime number having digit sum 8 is 3/25

From the Deck of 52 cards, if a card is randomly chosen, find the probability of getting a card with (i) a prime number on it, (ii) face on it. (M'18)

Step 1: Understand deck composition

Total cards = 52
Number cards: 2, 3, 4, 5, 6, 7, 8, 9, 10 (9 per suit × 4 suits = 36)
Face cards: Jack, Queen, King (3 per suit × 4 suits = 12)
Aces: 1 per suit × 4 suits = 4

(i) Probability of prime number card

Step 2: Find prime number cards

Prime numbers: 2, 3, 5, 7 (4 per suit)
Total prime number cards = 4 × 4 = 16
P(prime) = 16/52 = 4/13

(ii) Probability of face card

Step 3: Calculate probability

Face cards = 12
P(face) = 12/52 = 3/13

∴ (i) P(prime) = 4/13, (ii) P(face) = 3/13

If two dice are thrown at the same time, find the probability of getting sum of the dots on top is prime. (M'19)

Step 1: Find total outcomes

When two dice are thrown:
Total outcomes = 6 × 6 = 36

Step 2: Find prime sums possible

Possible sums: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
Prime sums: 2, 3, 5, 7, 11

Step 3: Find favorable outcomes for each prime sum

Sum = 2: (1,1) → 1 outcome
Sum = 3: (1,2), (2,1) → 2 outcomes
Sum = 5: (1,4), (2,3), (3,2), (4,1) → 4 outcomes
Sum = 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) → 6 outcomes
Sum = 11: (5,6), (6,5) → 2 outcomes
Total favorable outcomes = 1 + 2 + 4 + 6 + 2 = 15

Step 4: Calculate probability

P(prime sum) = 15/36 = 5/12

∴ The probability of getting a prime sum is 5/12

From a pack of 52 playing cards, Jacks, Queens, Kings and Aces of red colour are removed. From the remaining, a card is drawn at random. Find the probability that the card drawn is (i) a black queen, (ii) a red card. (J'19)

Step 1: Find cards removed

Red Jacks = 2, Red Queens = 2, Red Kings = 2, Red Aces = 2
Total cards removed = 8
Cards remaining = 52 - 8 = 44

(i) Probability of black queen

Step 2: Find black queens remaining

Black queens = 2 (not removed)
P(black queen) = 2/44 = 1/22

(ii) Probability of red card

Step 3: Find red cards remaining

Original red cards = 26
Red cards removed = 8
Red cards remaining = 26 - 8 = 18
P(red card) = 18/44 = 9/22

∴ (i) P(black queen) = 1/22, (ii) P(red card) = 9/22

A box contains 20 cards which are numbered from 1 to 20. If one card is selected at random from the box, find the probability that it bears (i) a prime number, (ii) an even number. (May 2022)

Step 1: Find total outcomes

Total cards = 20

(i) Probability of prime number

Step 2: Find prime numbers between 1 and 20

Prime numbers: 2, 3, 5, 7, 11, 13, 17, 19
Total prime numbers = 8
P(prime) = 8/20 = 2/5

(ii) Probability of even number

Step 3: Find even numbers between 1 and 20

Even numbers: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20
Total even numbers = 10
P(even) = 10/20 = 1/2

∴ (i) P(prime) = 2/5, (ii) P(even) = 1/2

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zaheerpashamd

M.Sc,B.Ed: SDVR ZPHS B.Gangaram Sathupally(md) Khammam(dt)
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