Q1. A teacher mixes a large bag of sweets of different colours and randomly selects a sample of 30 sweets. She counts the number of sweets of each colour: (i) Calculate the probability that a randomly picked sweet from the sample is green. (ii) If there are 600 sweets in total in the large bag, estimate how many are likely to be yellow, based on the sample results.

• 10 red sweets
• 8 green sweets
• 7 yellow sweets
• 5 blue sweets

Answer:

(i) Probability of picking a green sweet:

Number of green sweets = 8
Total sweets in sample = 30

P(green) = Number of favourable outcomes / Total number of trials
P(green) = 8/30 = 4/15 ≈ 0.267 or 26.7%

(ii) Estimate of yellow sweets in the bag:

From the sample, 7 out of 30 sweets are yellow.
Relative frequency of yellow = 7/30

Estimated number of yellow sweets in 600 = (7/30) × 600 = 7 × 20 = 140 sweets

So, approximately 140 sweets are likely to be yellow.

Q2. A survey is conducted at a school where a random sample of 40 students is asked about their favourite club. The responses are: Assume there are 800 students in the whole school. (i) What is the probability that a randomly chosen student from the sample prefers the Arts Club? (ii) Using the sample results, estimate how many students in the whole school are likely to prefer the Sports Club.

• 14 students: Science Club
• 11 students: Arts Club
• 9 students: Sports Club
• 6 students: Debate Club

Answer:

(i) Probability of preferring Arts Club:

Number of students who like Arts Club = 11
Total students in sample =14+11+9+6= 40

P(Arts Club) =Number of Students likeArts/ total Students

                    = 11/40 = 0.275 or 27.5%

(ii) Estimate of Sports Club preference in the whole school:

From the sample, 9 out of 40 students prefer Sports Club.

so P(Sports Club)= 9/40

Estimated number of students prefer Sports club in 800 = (9/40) × 800 = 9 × 20 = 180 students

So, approximately 180 students in the whole school are likely to prefer the Sports Club.

Q3. Toss a coin 20 times and record the result each time (heads or tails). (i) How many times did you get heads? (ii) How many times did you get tails? (iii) Calculate the experimental probability of getting heads. (iv) If you toss the coin once more, what is the probability of getting tails?

Answer:

Note: Since this is a hands-on activity, the actual numbers will vary. A sample solution is provided below.

(i) and (ii): Suppose in your 20 tosses, you got Heads 12 times and Tails 8 times.

(iii) Experimental probability of Heads:
P(Heads) = Number of times Heads occurred / Total tosses
P(Heads) = 12/20 = 3/5 = 0.6 or 60%

(iv) Probability of getting tails on the next toss:
A fair coin has no memory. The probability of getting tails on any single toss is always 1/2 = 0.5 or 50%, regardless of previous results.

Q4. Toss a paper cup into the air 100 times. After each toss record whether the cup lands on its bottom, upside down on its top or on its side (See Fig. 7.5). Assign probabilities to the outcomes by using experimental probability.                                          

Answer:

Note: This is an activity, so the actual probabilities depend on your results. A sample is shown below.

Suppose after 100 tosses, your results are:

• Bottom: 30 times
• Top (upside down): 20 times
• Side: 50 times

Experimental probabilities:

• P(Bottom) = number of times it landed on bottom/total number of tosses

=30/100

= 0.3 or 30%

• P(Top) =number of times it landed on Top/total number of tosses

 =20/100

 = 0.2 or 20%

• P(Side) =number of times it landed on Side/total number of tosses

= 50/100

= 0.5 or 50%

These probabilities are based on your actual observations and may differ from others' results.

Q5. What is the probability of getting an even number when rolling a fair 6-sided die?

Answer:

When rolling a standard 6-sided die, the possible outcomes are,

Sample space : S={1, 2, 3, 4, 5, 6}.

Even numbers on a die = {2, 4, 6} → 3 favourable outcomes.

Total possible outcomes = 6.

P(getting an even number) = Number of favourable outcomes / Total possible outcomes

P(even) = 3/6 = 1/2 = 0.5 or 50%.

So, there is an equal chance (50%) of rolling an even number.

Q6. Suppose you roll a 6-sided die 12 times and get a '3' three times. (i) What is the experimental probability of rolling a '3'? (ii) What is the theoretical probability of rolling a '3'? (iii) Why might these probabilities be different? What would you expect to happen if you roll the die 60, 600, or 6000 times?

Answer:

(i) Experimental Probability of rolling a '3':

Number of times 3 occurred = 3
Total rolls = 12

P(experimental) =P(3)= Number of times 3 Occurs / Total Roll

                                     =3/12 = 1/4 = 0.25 or 25%

(ii) Theoretical Probability:

For a fair die, there is 1 favourable outcome (the number 3) out of 6 possible outcomes.

P(theoretical)= P(3)

                      = 1/6 ≈ 0.167 or 16.7%

So,the teaoretical probability of rolling a '3' is 1/6. 

(iii) Why are they different?

Experimental probability is based on a small number of trials (12 rolls), which may not perfectly reflect the theoretical probability due to chance. As the number of trials increases, the experimental probability tends to get closer to the theoretical probability. This is known as the Law of Large Numbers.

If you roll the die 60, 600, or 6000 times, the experimental probability of rolling a 3 will likely become closer and closer to 1/6 ≈ 0.167.