**1. The three events P, Q and R are as follows:**

**Event P: If an integer is chosen at random from 1 to 50, then the probability that the number is ‘divisible by 5’.**

**Event Q: A box contains 2 red, 3 black and 5 white balls. If a ball is drawn at random, then the probability that the ball drawn is a ‘red ball’.**

**Event R: English letters are arranged in a row. If a letter is chosen at random from the letters of the English alphabet, then the probability is that it is a letter of the word ‘JAGUAR’.**

**Which of the above events have probabilities equal to 0.2?**

a) Both events P and Q

b) Both events Q and R

c) Both events R and P

d) All events P, Q and R

**Answer:** a) Both events P and Q

**Explanation:**

**Event P:**

Total number of outcomes = 50

Number of favourable outcomes {5, 10, 15, 20, 25, 30, 35, 40, 45, 50} = 10

Required Probability = 10/ 50 = 1/5 = 0.2

**Event Q:**

Number of red balls = 2

Number of black balls = 3

Number of white balls = 5

Total number of balls = 2 + 3 + 5 = 10

Probability of getting a red ball = 2/10 = 1/5 = 0.2

**Event R:**

Total number of English alphabets = 26

Letters of JAGUAR {A, G, J, U, R} = 5

Required Probability = 5/26 = 0.19

∴ Both events P and Q have probabilities equal to 0.2.

**2. A bag contains some blue balls and 45 red balls. If the probability of drawing a blue ball is three-fifths of a red ball, then what is the number of blue balls in the bag?**

a) 21

b) 23

c) 27

d) 29

**Answer:** c) 27

**Explanation: **Let the bag contain x blue balls.

Number of red balls = 45

Total number of balls in a bag = x+45

Probability of drawing a blue ball = ^{x}⁄_{x+45}

Probability of drawing a red ball = ^{45}⁄_{x+45}

According to the question,

Probability of drawing a blue ball is three-fifths of a red ball.

⇒ ^{x}⁄_{x+45 }= ^{3}⁄_{5} × ^{45}⁄_{x+45}⇒ x = ^{3×45}⁄_{5}∴ x = 27

**3. Cards are labelled as c, d, e,....., s, t. They are put in a box and shuffled. A student is asked to draw a card from the box. What is the probability that the card draws none of the letters of the word ‘jacket’?**

a) 33

b) 53

c) 67

d) 87

**Answer:** c) 0.67

**Explanation: **S = {c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t}

Total number of cards = n(S) = 18

E = {j, a, c, k, e, t} = {a, c, e, k, j, t}

Number of the letters in the word ‘jacket’ = n(E) = 6

Number of letters not in the word ‘jacket’ = n(E′) = n(S) − n(E) = 18 − 6 = 12

Probability that the card draws none of the letters of the word ‘jacket’

= ^{n(E′)}⁄_{n(S)} = ^{12}⁄_{18} = 0.67

**4. The events are as follows:**

**Event C: If a card is selected at random from a pack of 52 cards, then a blackface card is found.**

**Event D: If a dice is thrown, then an odd number more than 2 is found on the top of a dice.**

**Event T: If a ticket is drawn at random from a box containing tickets numbered 1 to 25, then the selected ticket has a number which is a multiple of 7.**

**Which of the above events have probabilities equal to 1/3?**

a) Both events C and D

b) Both events D and T

c) Both events T and C

d) All events C, D and T

**Answer:** b) Both events D and T

**Explanation: **

**Event C:** Total number of cards = 52

Number of black face cards = 3 × 2 = 6

Required Probability = 6/52 = 3/26

**Event D: **A die is thrown.

Total number of events = 6

Total number of odd numbers more than 2 = 2 (i.e., 3, 5)

Required Probability = 2/6 = 1/3

**Event T: **There are a total 25 tickets in a bag.

Number of tickets which is multiple of 7 = 3 (7, 14 and 21)

Required Probability = ^{3}⁄_{15} = 1/3

Both events D and T have probabilities equal to 1/3.

**5. What is the probability when two dice are thrown simultaneously whose sum is at most 7?**

a) 36/42

b) 36/49

c) 49/64

d) 49/84

**Answer:** d) 49/84

**Explanation: **The sum includes numbers that are both less than 7 and equal to 7. The sum whose at most is 7 is marked as

Total Outcomes = n(s) = 36

Favourable Outcomes = n(E) = 21

Probability when two dice are thrown simultaneously whose sum is at most 7.

P(E) = n(E)/n(s) = 21/36 = 7/12

7/12 = 49/84

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