Probability is primarily a measure of:

1. A. length
2. B. likelihood of an event
3. C. area
4. D. number of outcomes only

Which of the following best describes a random experiment?

1. A. Its outcome is known in advance
2. B. It cannot be repeated
3. C. Its possible outcomes may be known but the actual outcome is uncertain
4. D. It always has two outcomes

The probability of an impossible event is:

1. A. −1
2. B. 0
3. C. 1/2
4. D. 1

The probability of a certain event is:

1. A. 0
2. B. 1/4
3. C. 1/2
4. D. 1

If P(E)=0.75, then E is:

1. A. impossible
2. B. less likely than not
3. C. more likely than not
4. D. certain

Which statement is correct for every event E?

1. A. P(E)<0
2. B. 0≤P(E)≤1
3. C. P(E)>1
4. D. P(E) is always rational

A die is rolled 50 times and a 4 appears 8 times. The experimental probability of 4 is:

1. A. 1/6
2. B. 4/25
3. C. 8/42
4. D. 42/50

Theoretical probability is computed directly from favourable/total outcomes only when:

1. A. outcomes are equally likely
2. B. there is experimental data
3. C. the sample is large
4. D. the event is certain

The sample space for tossing two fair coins is:

1. A. {H,T}
2. B. {HH,HT,TT}
3. C. {HH,HT,TH,TT}
4. D. {0,1,2}

When three coins are tossed and only the number of heads is recorded, a suitable sample space is:

1. A. {H,T}
2. B. {HHH,HHT,HTH,THH}
3. C. {0,1,2,3}
4. D. {1,2,3}

An event is:

1. A. always one outcome only
2. B. a subset of the sample space
3. C. the entire experiment
4. D. the number of trials

For S={1,2,3,4,5,6}, the event 'number greater than 4' is:

1. A. {4,5,6}
2. B. {5,6}
3. C. {1,2,3,4}
4. D. {6}

A tree diagram is most useful for:

1. A. measuring length
2. B. listing outcomes of multi-step experiments
3. C. showing only impossible events
4. D. finding experimental frequency without trials

Two fair coin tosses give P(HH)=

1. A. 1/2
2. B. 1/3
3. C. 1/4
4. D. 3/4

For two fair coin tosses, P(exactly one head)=

1. A. 1/4
2. B. 1/2
3. C. 3/4
4. D. 1

A fair die shows six 6s in a row. P(6 on the next roll) is:

1. A. 0
2. B. 1/36
3. C. 1/6
4. D. greater than 1/6

Believing that tails is 'due' after six consecutive heads is called:

1. A. sampling bias
2. B. Gambler's Fallacy
3. C. relative frequency
4. D. sample-space error

As the number of fair repeated trials becomes large, experimental probability tends to:

1. A. 0
2. B. 1
3. C. the theoretical probability
4. D. become impossible to calculate

A fair coin is called unbiased because:

1. A. it always alternates H,T
2. B. neither face has a reason to occur more often
3. C. it has no mass
4. D. it cannot land on an edge

A sample has 20 mango-lovers among 50 students. The relative frequency is:

1. A. 0.2
2. B. 0.4
3. C. 0.5
4. D. 2.5

Using the 0.4 sample estimate for a school of 1500, the estimated number preferring mango is:

1. A. 400
2. B. 500
3. C. 600
4. D. 750

A box has 5 green and 7 red balls. If one ball is drawn and only colour is recorded, the sample space is:

1. A. {G,R}
2. B. {G,G,G,G,G,R,…}
3. C. {1,…,12}
4. D. {5,7}

A bag has 3 red and 7 blue marbles. Which statement is correct?

1. A. Red and blue are equally likely because there are two colours
2. B. P(red)=3/10 and P(blue)=7/10
3. C. P(red)=1/2
4. D. Theoretical probability cannot be found

The letters of PEACE are on five identical cards. P(not E) is:

1. A. 1/5
2. B. 2/5
3. C. 3/5
4. D. 4/5

A spinner has equally likely outcomes 1 to 8. P(multiple of 3) is:

1. A. 1/8
2. B. 1/4
3. C. 3/8
4. D. 1/2

Two dice are rolled. The probability that the sum is 7 is:

1. A. 1/12
2. B. 1/9
3. C. 1/6
4. D. 5/36

Two balls are drawn without replacement from 4 red and 5 blue. P(red then blue) is:

1. A. 4/9×5/9
2. B. 4/9×5/8
3. C. 5/9×4/8
4. D. 20/81

From the same basket, P(two blue) is:

1. A. 5/9×5/9
2. B. 5/9×4/8
3. C. 5/8×4/7
4. D. 4/9

A dye is dropped uniformly at random on a 3 m by 2 m rectangle containing a circle of diameter 1 m. The probability it lands in the circle is:

1. A. π/24
2. B. π/12
3. C. π/6
4. D. 1/6

Three multiple-choice questions have 4 options each, one correct. A student guesses. P(exactly 2 correct) is:

1. A. 3/64
2. B. 9/64
3. C. 27/64
4. D. 1/16

Define probability in one sentence.

What is randomness?

Write the probability scale.

What is a sample space?

What is an event?

Write the sample space for one fair coin toss.

Write the sample space for one standard die roll.

Write the sample space for two coin tosses.

Write the experimental probability formula.

Write the theoretical probability formula for equally likely outcomes.

A die is rolled 12 times and a 3 appears 3 times. Find the experimental probability.

For a fair die, find P(3).

A fair coin gives 6 heads in succession. Find P(tail on the next toss).

In a sample of 50 students, 15 like football. Find the relative frequency.

If S={HH,HT,TH,TT}, write the event 'at least one head'.

Differentiate experimental probability from theoretical probability.

A teacher samples 30 sweets: 10 red, 8 green, 7 yellow, 5 blue. Find the experimental probability of green.

Using the same sample, estimate how many yellow sweets there are in a bag of 600 sweets.

A sample of 40 students has 11 Arts Club preferences. Find the estimated probability and expected count in a school of 800.

A fair die is rolled. Find P(even number).

A die is rolled. Find P(number greater than 4).

Two coins are tossed. Find P(at least one head).

Three coins are tossed. Find P(exactly two heads).

Ten cards numbered 1 to 10 are equally likely. Find P(even card).

A bag contains 3 red,2 blue,1 green ball. Find P(not red).

A random letter is chosen from PROBABILITY. Find P(B).

The letters of PEACE are on identical cards. Find P(P,E or C).

A spinner has equally likely numbers 1 to 8. Find P(odd) and P(number>2).

Explain why getting a 6 three times in a row does not change P(6) on the next fair die roll.

A box has 5 green and 7 red balls. Explain why S={G,R} does not mean P(G)=P(R).

Explain the Law of Large Numbers in the context of rolling a fair die.

A die is rolled 12 times and a 3 appears 3 times. Compare experimental and theoretical probabilities and explain the difference.

Construct the sample space for rolling one die and tossing one coin together, and find P(even number and H).

A village fair offers 3 snacks (Samosa,Pakora,Bhaji) and 2 drinks (Chai,Lassi). List the sample space and find P(Samosa) if each combination is equally likely.

Two fair coins are tossed. Use a tree-diagram argument to find P(one head and one tail).

Basket A has one apple and two oranges; Basket B has one banana and one mango. One fruit is selected randomly from each basket. Find P(apple and banana).

A box has 3 red,4 black,2 green pens. A pen is picked, replaced, and another is picked. Find P(same colour).

A tyre company's 1000 records are 20 (<4000), 210 (4001–9000), 325 (9001–14000),445 (>14000). Find the three probabilities requested in the chapter.

A basket has 4 red and 5 blue balls. Two are drawn without replacement. Find P(R then B) and P(BB).

Two fair dice are rolled. Find P(sum is a prime greater than 5).

A bag has 4 red,3 green,2 blue balls. Two are drawn without replacement. Find P(different colours).

Three fair coins are tossed. Find P(first coin is H and exactly two heads occur in total).

A four-digit number is formed using 1,2,3,4 without repetition. Find P(number is even).

A student guesses answers to 3 questions, each with 4 options. Find P(exactly 2 correct).

A dye is dropped uniformly on a 3m×2m rectangle. A circle of diameter 1m lies inside it. Find the probability of landing in the circle.

A box has balls numbered 1 to 4. Two draws are made with replacement. Write the sample space and its size; then find P(the two numbers are different).

Repeat the previous experiment without replacement. Write the sample space size and find P(the second number is greater than the first).

A bag contains 4 red and 5 blue balls. Two are drawn without replacement. Find P(one of each colour), P(same colour), and verify the probabilities sum to 1.

Two fair dice are rolled. Find the probability that the sum is a prime greater than 5 and compare it with the probability that the sum is 7.

A fair coin is tossed 8 times and gives 8 heads. A student says tails is now more likely. Identify the fallacy and give the correct next-toss probability. Then distinguish this from the probability of getting 9 heads in a row from the start.

A biased-looking bag contains 3 red and 7 blue marbles. Explain why the colour outcomes are not equally likely, yet individual marble outcomes may be considered equally likely. Use this to find P(red).

A random sample of 50 students gives mango preference 20, apple 15, banana 10, grapes 5. Estimate counts for a school of 1500 and explain one limitation.

For three fair coins, compare these two experiments: A records the full sequence; B records only number of heads. Write each sample space and explain why the four outcomes of B are not equally likely.

A spinner has equally likely labels 1 to 8. Find P(odd), P(>2), P(<9), P(multiple of 3), and explain which event is certain.

A rectangle is 3m×2m and contains a circle of diameter 1m. If a point is selected uniformly at random, derive the geometric probability of landing inside the circle and outside the circle.