### Q3. Probability and Its Importance:
#### (a) What is Probability?
Probability is a measure of the likelihood that an event will occur. It quantifies the uncertainty and is expressed as a number between 0 and 1, where 0 indicates an impossible event and 1 indicates a certain event.
Importance in Social and Economic Lives:
1. Decision Making: Probability helps in making informed decisions in uncertain situations such as investments, policy-making, and insurance.
2. Risk Management: It aids in assessing and managing risk by estimating the likelihood of various outcomes.
3. Predictive Analysis: In economics and social sciences, probability is used to predict future trends, such as market behavior, weather forecasting, and public health trends.
4. Quality Control: It is used in manufacturing processes to maintain and improve the quality of products.
#### (b) Experiment of Tossing a Pair of Fair Dice:
(i) Probability of Getting a Sum of Six:
The sample space of tossing two dice consists of 36 possible outcomes. The favorable outcomes for a sum of six are (1,5), (2,4), (3,3), (4,2), and (5,1), which are 5 favorable outcomes.
[tex]\[ P(\text{sum of six}) = \frac{5}{36} \][/tex]
(ii) Probability of Getting an Even Number:
An even number on the dice can be 2, 4, or 6. Each die has 3 even numbers, so the combinations of getting an even number in at least one of the dice:
[tex]\[ 3/6 \times 3/6 = 0.25 \][/tex]
(iii) Probability of Getting a Sum of Six or Ten:
The favorable outcomes for a sum of ten are (4,6), (5,5), and (6,4), which are 3 outcomes.
[tex]\[ P(\text{sum of six or ten}) = \frac{5 + 3}{36} = \frac{8}{36} = \frac{2}{9} \][/tex]
(iv) Probability of Getting Coincident Numbers (Same Number on Both Dice):
The favorable outcomes for coincident numbers are (1,1), (2,2), (3,3), (4,4), (5,5), and (6,6), which are 6 outcomes.
[tex]\[ P(\text{coincident numbers}) = \frac{6}{36} = \frac{1}{6} \][/tex]
### Q4. Probability of Machines Breaking Down:
The problem follows a binomial distribution with parameters [tex]\( n = 5 \)[/tex] and [tex]\( p = 0.25 \)[/tex].
(a) Probability that No Machine Breaks Down:
[tex]\[ P(X = 0) = \binom{5}{0} (0.25)^0 (0.75)^5 = (1)(0.75)^5 = 0.2373 \][/tex]
(b) Probability that Exactly One Machine Breaks Down:
[tex]\[ P(X = 1) = \binom{5}{1} (0.25)^1 (0.75)^4 = 5 \times 0.25 \times 0.3164 = 0.3955 \][/tex]
(c) Probability that At Least Two Machines Break Down:
We calculate the probability of 0 or 1 machine breaking down and subtract from 1:
[tex]\[ P(X \geq 2) = 1 - (P(X = 0) + P(X = 1)) \][/tex]
[tex]\[ P(X \geq 2) = 1 - (0.2373 + 0.3955) = 1 - 0.6328 = 0.3672 \][/tex]
(d) Probability that All Machines Break Down:
[tex]\[ P(X = 5) = \binom{5}{5} (0.25)^5 (0.75)^0 = (1)(0.25)^5 = 0.00098 \][/tex]
### Q5. Distinguishing Between Concepts:
#### (a) Random Sampling and Cluster Sampling:
- Random Sampling: Involves selecting individuals from a population in such a way that every individual has an equal chance of being chosen. It is used to ensure unbiased representation.
- Cluster Sampling: Involves dividing a population into clusters (groups), then randomly selecting some of these clusters. All individuals within the chosen clusters are surveyed. It is often used for efficiency and cost reduction in large populations.
#### (b) Population:
Assuming the term 'Populatis' is a typographical error referring to 'Population.'
- Population: Refers to the entire group of individuals or items that one wants to study or draw conclusions about. In statistics, a population is considered the complete set of items or events being examined.
