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**Question 1. If we throw a die, then the upper face shows 1 or 2; or 3 or 4; or 5 or 6. Suppose we throw a die 150 times and get 2 for 75 times. What is the probability of getting a â€˜2â€™? Solution: **Let E be the event of getting 2.

âˆ´ P(E) = Number of favourable outcomes/ total number of trails

= (75/150) = (1/2) = 0.5

**Question 2. A coin is tossed 200 times and is found that a tail comes up for 120 times. Find the probability of getting a tail. Solution: **Number of trials = 200

Number of favourable outcomes = Number of getting a tail

= 120

Let the probability of getting a tail is P(E).

âˆ´ P(E) = Number of favourable outcomes/ total number of trails = (120/200) = (6/10) = 0.6

**Question 3. If a coin is tossed for a certain number of times. How many times the coin was tossed, if the probability of getting a head is 0.4 and it appeared up for 24 times? Solution:** Number of favourable outcomes = 24

Let the total number of trials is n.

Here, P(E) = Number of favourable outcomes/ total number of trails

.

**Question 4. In a cricket match, if the probability P(E) of hitting the boundary is 0.3, then find the probability of not-hitting the boundary. Solution: **Probability of hitting the boundary P(E) = 0.3

âˆ´ Probability of not-hitting the boundary = 1 â€“ P(E) = 1 â€“ 0.3 = 0.7

**Question 5. In a GK test a student was given 50 questions one by one. He gave the correct answer for 30 questions. Find the probability of giving correct answers. Solution: **Total number of trials = 50

Number of favourable outcomes = Number of correct answers = 30

Let P(E) be the probability of giving correct answers.

âˆ´ P(E) = Number of favourable outcomes/ total number of trails = (30/50) = (60/100) = 0.6

**Question 1. Write â€˜trueâ€™, â€˜falseâ€™ or â€˜ambiguousâ€™ for each of the following statements: (i) All prime numbers are odd. (ii) Division by 0 is not possible. (iii) There are 8 days in a week. (iv) 2 + 3 = 5 (v) Raju is a poor boy. (vi) Cats can fly. Solution:** (i) False (ii) True (iii) False (iv) True (v) Ambiguous (vi) False

**Question 2. State whether the following statements are true or false: (i) The sum of the interior angles of a triangle is 360Â° (ii) The number 2 is the only even prime number. (iii) Every odd number is greater than 2. (iv) Every angle formed in semicircle is 90Â°. (v) When we divide a number by 0, we get 1. (vi) For every real number x, x ^{2} = 2x. Solution:** (i) False (ii) True (iii) False (iv ) True (v) False (vi ) False

**Question 3. Define a (i) Theorem, (ii) Axiom and (iii) Conjecture. Solution:**

**Question 4. Prove that the product of two even natural numbers is divisible by 16. Solution:** Let the two even numbers be x and y.

Since, x is an even number, it is divisible by 2.

âˆ´ x = 2m [where m is a natural number]

Also, y = 2n [where n is a natural number]

âˆ´ xy = (2m) x (2n) = 4mn Since, 4mn is divisible by 2.

âˆ´ xy is also divisible by 2.

Thus, xy is even.

**Question 5. A coin is tossed 150 times and it is found that head comes 115 times and tail 35 times. If a coin tossed at random, what is the probability of getting (i) a head (ii) a tail Solution:** Here, total number of trials = 150

(i) âˆµ Number of heads as outcome = 115

âˆ´ Probability of an event of getting a head = (115/150)= (23/30)

(ii) âˆµ Number of tails as outcomes = 35

âˆ´ Probability of an event of getting a tail = (35/150) = (7/30)

**REMEMBER**

(i) The probability of an event can be 0 to 1.

(ii) [Probability of the occurrence of an event] + [Probability of non-occurrence of that event] = 1

(iii) The sum of the probability of all the possible outcomes of a trial = 1

**Question 6. A die is thrown 270 times and the outcomes are recorded as in the following table:**

Outcome | 1 | 2 | 3 | 4 | 5 | 6 |

Frequency | 36 | 45 | 33 | 18 | 75 | 63 |

**If a die is thrown at random, find the probability of getting: (i) 1 (ii) 2 (iii) 3 (iv) 4 (v) 5 (vi) 6 Solution: **Here, total number of trials = 270

Let P(E

Now, (i) âˆµ Number of events of getting 1 = 36

âˆ´ Probability of a throw to get 1

(ii) âˆµ Number of events of getting 2 = 45

âˆ´ Probability of a throw to get 2

(iii) âˆµ Number of events of getting 3 = 33

âˆ´ Probability of a throw to get 3

(iv) âˆµ Number of events of getting 4 = 18

âˆ´ Probability of a throw to get 4

(v) âˆµ Number of events of getting 5 = 75

âˆ´ Probability of a throw to get 5

(vi) âˆµ Number of events of getting 6 = 63

âˆ´ Probability of a throw to get 6

Note: P(E_{1}) + P(E_{2}) + P(E_{3}) + P(E_{4}) + P(E_{5}) + P(E_{6})

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