Probability

(i) 1

(ii) 0; such an event is called an impossible event.

(iii) 1; such an event is called a sure event or certain event.

(iv) 1

(v) greater than or equal to 0 and less than or equal to 1.

None of these experiments is guaranteed in the chapter to have equally likely outcomes.

- (i) A car may or may not start; these outcomes are not necessarily equally likely.
- (ii) A basketball player may score or miss; these are not equally likely in general.
- (iii) A true-false question has right or wrong as outcomes, but they are not necessarily equally likely.
- (iv) A baby being a boy or a girl is not treated here as equally likely for probability calculations.

So, none of the four experiments has equally likely outcomes in the sense required for the theoretical probability used in this chapter.

Tossing a coin is considered fair because the coin is assumed to be unbiased and the toss is random. So the two outcomes, head and tail, are equally likely. Therefore, neither team has an advantage, and the result is fair.

The probability of an event must satisfy $0 \le P(E) \le 1$.

- $\frac{2}{3}$ is valid.
- **$-1.5$** is not valid because probability cannot be negative.
- $15\% = 0.15$ is valid.
- $0.7$ is valid.

So the value that cannot be a probability is -1.5.

For complementary events,
$$P(\text{not }E)=1-P(E)$$
Given $P(E)=0.05$,
$$P(\text{not }E)=1-0.05=0.95$$

The bag contains only lemon-flavoured candies.

(i) An orange flavoured candy cannot be taken out, so the probability is
$$0$$

(ii) A lemon flavoured candy is certain to be taken out, so the probability is
$$1$$

If the probability that the 2 students do not have the same birthday is $0.992$, then the probability that they have the same birthday is the complement:
$$1-0.992=0.008$$

Total balls $=3+5=8$.

(i) Probability of red:
$$P(\text{red})=\frac{3}{8}$$

(ii) Probability of not red means probability of a black ball:
$$P(\text{not red})=\frac{5}{8}$$

Total marbles $=5+8+4=17$.

(i) Red marbles $=5$,
$$P(\text{red})=\frac{5}{17}$$

(ii) White marbles $=8$,
$$P(\text{white})=\frac{8}{17}$$

(iii) Not green means red or white, so favourable marbles $=5+8=13$,
$$P(\text{not green})=\frac{13}{17}$$

Total coins $=100+50+20+10=180$.

(i) Probability of a 50 p coin:
$$\frac{100}{180}=\frac{5}{9}$$
(ii) Probability of not a ₹5 coin:
$$\frac{180-10}{180}=\frac{170}{180}=\frac{17}{18}$$

So the computed answers are **$\frac{5}{9}$ and $\frac{17}{18}$**. These are not among the printed options because the book does not give options for this numerical question.

Total fish $=5+8=13$.

Favourable outcomes for a male fish $=5$.

So,
$$P(\text{male fish})=\frac{5}{13}$$

There are 8 equally likely outcomes: $1,2,3,4,5,6,7,8$.

(i) Pointing at 8: favourable outcomes $=1$,
$$P=\frac{1}{8}$$

(ii) Pointing at an odd number: $1,3,5,7$ so favourable outcomes $=4$,
$$P=\frac{4}{8}=\frac{1}{2}$$

(iii) Pointing at a number greater than 2: $3,4,5,6,7,8$ so favourable outcomes $=6$,
$$P=\frac{6}{8}=\frac{3}{4}$$

(iv) Pointing at a number less than 9: all 8 outcomes are favourable,
$$P=\frac{8}{8}=1$$

A die has outcomes $1,2,3,4,5,6$.

(i) Prime numbers are $2,3,5$ so favourable outcomes $=3$:
$$P=\frac{3}{6}=\frac{1}{2}$$

(ii) Numbers lying between 2 and 6 are $3,4,5$ so favourable outcomes $=3$:
$$P=\frac{3}{6}=\frac{1}{2}$$

(iii) Odd numbers are $1,3,5$ so favourable outcomes $=3$:
$$P=\frac{3}{6}=\frac{1}{2}$$

These are the standard answers from the chapter context. If strictly following the chapter’s examples, the result for each is $\frac{1}{2}$, and that is the computed value.