NCERT Solutions for Class 9 Maths Chapter 7 — Exercise Set 7.2#
$$\text{Experimental probability} = \dfrac{\text{Number of times the event occurred}}{\text{Total number of trials}}$$$$\text{Theoretical probability} = \dfrac{\text{Number of favourable outcomes}}{\text{Number of possible outcomes}}$$
To estimate a total from a sample, multiply the sample probability by the size of the whole group.
1. 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: 10 red sweets | 8 green sweets | 7 yellow sweets | 5 blue sweets #
(i) Calculate the probability that a randomly picked sweet from the sample is green.
$$P(\text{green}) = \dfrac{4}{15}$$
∴ The probability is $\dfrac{4}{15}$ ≈ 0.27.
(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.
In the sample:
$$P(\text{yellow}) = \dfrac{7}{30}$$For the whole bag of 600 sweets:
$$\text{Yellow sweets} = \dfrac{7}{30} \times 600$$$$\text{Yellow sweets} = 140$$
∴ About 140 sweets are likely to be yellow.
2. A survey is conducted at a school where a random sample of 40 students is asked about their favourite club. The responses are: 14 students: Science Club | 11 students: Arts Club | 9 students: Sports Club | 6 students: Debate Club. 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?
∴ The probability is $\dfrac{11}{40}$ = 0.275.
(ii) Using the sample results, estimate how many students in the whole school are likely to prefer the Sports Club.
$$\text{Students} = 180$$
∴ About 180 students are likely to prefer the Sports Club.
3. Toss a coin 20 times and record the result each time (heads or tails). #
This is an experiment, so your numbers will differ. Here is one sample record to show the method.
Results of 20 tosses, in order:
H T T H H T H T H H T H T T H H T H H T
(i) How many times did you get heads?
Counting the H’s in the sample record: 11 heads.
(ii) How many times did you get tails?
So 9 tails.
(iii) Calculate the experimental probability of getting heads.
$$P(\text{heads}) = 0.55$$
With your own record, use $\dfrac{\text{your number of heads}}{20}$.
(iv) If you toss the coin once more, what is the probability of getting tails?
Each toss is a fresh start: the coin does not remember earlier results. A fair coin has 2 equally likely outcomes, so:
$$P(\text{tails}) = \dfrac{1}{2}$$∴ The probability is $\dfrac{1}{2}$, whatever happened in the first 20 tosses.
4. 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. #
Solution
This is an experiment, so results vary. A cup is not symmetrical, so the three outcomes are not equally likely, and the probabilities must come from the experiment. One sample record:
| Outcome | Bottom | Top (upside down) | Side | Total |
|---|---|---|---|---|
| Number of times | 18 | 23 | 59 | 100 |
| Experimental probability | $\dfrac{18}{100} = 0.18$ | $\dfrac{23}{100} = 0.23$ | $\dfrac{59}{100} = 0.59$ | 1 |
The probabilities always add up to 1, because every toss has exactly one of the three outcomes. Landing on its side is usually the most likely.
5. What is the probability of getting an even number when rolling a fair 6-sided die? #
Solution
The possible outcomes are 1, 2, 3, 4, 5, 6, all equally likely. The favourable outcomes are 2, 4, 6:
$$P(\text{even}) = \dfrac{3}{6}$$$$P(\text{even}) = \dfrac{1}{2}$$
∴ The probability is $\dfrac{1}{2}$.
6. 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’?
$$P(3) = \dfrac{1}{4}$$
∴ The experimental probability is $\dfrac{1}{4}$ = 0.25.
(ii) What is the theoretical probability of rolling a ‘3’?
One favourable outcome out of 6 equally likely outcomes:
$$P(3) = \dfrac{1}{6}$$∴ The theoretical probability is $\dfrac{1}{6}$ ≈ 0.17.
(iii) Why might these probabilities be different? What would you expect to happen if you roll the die 60, 600, or 6000 times?
Why different: 12 rolls is a very small number of trials. Chance causes big swings in small samples — getting one extra ‘3’ changes the result a lot.
More rolls: as the number of rolls grows, the experimental probability gets closer to $\dfrac{1}{6}$. We would expect about
| Rolls | Expected number of 3s |
|---|---|
| 60 | about 10 |
| 600 | about 100 |
| 6000 | about 1000 |
The actual counts will still vary a little, but the relative frequency settles down near $\dfrac{1}{6}$ as the number of trials increases.