Think and Reflect (Page no 156)
Q1. Such unpredictability can be useful sometimes! For example, in a cricket match, the fact that a coin is tossed to decide which team will bat first is considered to be a fair method. Can you explain why?
Answer:
Tossing a coin is considered a fair method because it is a random experiment with two equally likely outcomes — Heads or Tails.
Probability of Heads = 1/2 (50%)
Probability of Tails = 1/2 (50%)
Since neither outcome is favoured, each team has an equal chance of winning the toss. This eliminates any bias or unfair advantage in deciding who bats first.
Qpage no 157. Ask your friend to predict the outcome of a ₹1 coin you toss. Do you see that your friend could guess heads or tails but could not know for certain? That's randomness! All possible results are known, but each individual try is unpredictable.
Answer:
This activity demonstrates the concept of randomness. Although the possible outcomes (Heads or Tails) are known in advance, the result of a single toss cannot be predicted with certainty.
for example:
Coin toss P(H) = 1/2, P(T) = 1/2
Each toss is independent and the outcome is determined by chance.
Q3. Excercise Set 7.1 Rank the following events on a scale from 0 (Impossible) to 1 (Certain). Label each event: Impossible, less likely, equally likely (even chance), more likely, certain. Give reasons why you gave each event its ranking. (i) The next Monday will come after Sunday. (ii) It will snow in Mumbai in July. (iii) An elephant will walk through your classroom today. (iv) You will greet at least one friend at school tomorrow.
Answer:
- (i) Certain (Probability = 1) — Monday always comes after Sunday; it is a fixed sequence in the week.
- (ii) Impossible (Probability = 0) — Mumbai has a tropical climate and July is a hot, rainy month; snowfall is impossible under normal conditions.
- (iii) Less likely (Probability close to 0) — It is highly unlikely that an elephant would walk into a classroom, though not strictly impossible.
- (iv) More likely (Probability close to 1) — In a school setting, it is very likely you will greet at least one friend, though not absolutely certain.
Q4. page no 163. If I have rolled a 4 on a die 8 times in succession, the probability of rolling a 4 again is still only ≈ 0.16 (assuming the die is fair). Probability does not tell you what will happen next but predicts what will happen in the long run.
Answer:
This reinforces the idea that each roll of a fair die is independent. Past outcomes do not affect future outcomes.
Sample Space (S) = {1, 2, 3, 4, 5, 6}, i.e., 6 outcomes
Suitable Outcome = {4}, i.e., 1 outcome
Probability: P(4) = 1/6 ≈0.16
The probability of rolling a 4 on any single roll is always 1/6 ≈ 0.167, regardless of previous results. Probability describes long-term patterns, not short-term sequences.
Q5. page 167. Think and Reflect When we used the sample space {Rain, No Rain} in Example 1, we focused only on whether it will rain or not. However, if we want to include different amounts of rainfall like drizzle, light rain or heavy rain, we need to expand the sample space to {No Rain, Drizzle, Light Rain, Heavy Rain} so that it better matches the level of detail required for the question. It is important to ensure the sample space is detailed enough to suit the specific problem being studied.
Answer:
The choice of sample space depends on the level of detail needed. A broader or more specific sample space helps in accurately describing the problem and calculating probabilities for different scenarios. For example, if we need to know the probability of any rainfall (regardless of intensity), the simpler sample space {Rain, No Rain} is sufficient. But if we want to distinguish between types of rain, we must use the expanded sample space {No Rain, Drizzle, Light Rain, Heavy Rain}.
Q6 page 169. Can you calculate the probability of getting one head and one tail?
Answer:
When tossing a fair coin twice, the sample space is S = {HH, HT, TH, TT}.
The outcomes with exactly one Head and one Tail are HT and TH.
So, number of favourable outcomes = 2, total outcomes = 4.
Therefore, P(one Head and one Tail) = favourable outcomes/Total outcomes
= 2/4 = 1/2 = 0.5 or 50%.