Probability and Statistics

In the world of mathematics, two subjects stand out for their wide range of applications in various fields: Probability and Statistics. These two disciplines are not just limited to theoretical concepts but have real-world significance that influences decision-making in industries, economics, science, and daily life. This blog will introduce you to the fundamental ideas of Probability and Statistics, illustrate their practical applications, along with examples to deepen your understanding.

1.0What is Probability?

Probability is a branch of mathematics focused on measuring the likelihood or chance of an event happening. It is a measure that helps us understand and predict the behavior of uncertain situations. Probability is represented by a value ranging from 0 to 1, where:

• 0 indicates that an event is impossible (it will not happen).
• 1 indicates that an event is certain (it will definitely happen).

For example, when flipping a fair coin, the probability of getting head is 0.5, or 50%, since there are two equally possible outcomes (heads or tails), and one of them is heads.

2.0Basic Concepts of Probability

1. Experiment: An action or process that leads to a set of possible outcomes. For example, tossing two coins is an experiment.
2. Outcome: The result of a single trial of an experiment. For instance, getting heads when tossing a coin.
3. Event: A set of outcomes from an experiment. If you toss two coins, getting one head and one tail is an event.
4. Sample Space: The set of all possible outcomes of an experiment. For a six-sided die, the sample space is {1, 2, 3, 4, 5, 6}.

3.0Probability Formula

The probability P(E) of an event E is calculated using the formula:

P(E)=$\frac{\text{ Number of favorable outcomes }}{\text{ Total number of outcomes }}$

Example:

If you roll a die, the probability of rolling a 4 is:

$P(\text{ rolling a }4)=\frac{1}{6}$

This is because there is only one favorable outcome (rolling a 4) and six possible outcomes in total (rolling a 1, 2, 3, 4, 5, or 6).

4.0What is Statistics?

Statistics is the discipline of gathering, analyzing, interpreting, and presenting data. It is a powerful tool that allows us to make sense of large amounts of information and draw conclusions based on that data. Statistics is divided into two main branches:

1. Descriptive Statistics: This involves summarizing and organizing data so that it can be easily understood. Common descriptive statistics include mean, median, mode, and standard deviation.
2. Inferential Statistics: This encompasses making predictions or drawing conclusions about a population from a sample of data, employing techniques such as hypothesis testing, confidence intervals, and regression analysis.

5.0Measure of Central Tendency

The three Main Measure of central tendency are

1. Mean
2. Mode 
3. Median

1. Mean:

Mean also known as the average the mean is calculated by summing up all the values in the data set and dividing them by the total number of values. It is sensitive to extreme values, making it useful for symmetric distributions.

Example:

For a given set,

3, 6, 9, 12, 15

The mean is calculated by summing all the numbers and dividing the total by the number of values.

$\text{ Mean }=\frac{3+6+9+12+15}{5}$

$\text{ Mean }=\frac{45}{5}$

Mean = 9

2. Median:

The median is determined as the central value within a dataset once the values have been arranged in either ascending or descending order. If the number of values is even, the median is calculated as the average of the two central values. It is less affected by extreme values and is useful for skewed distributions.

Example:

For a given set,

2, 5, 6, 8, 3, 9, 6, 7, 3, 10, 15

To find the median first arrange in ascending order.

2, 3, 3, 5, 6, 6, 7, 8, 9, 10, 15  

The median is determined by choosing the middle value which is 6.

3. Mode:

The mode is the value that appears most often in the data set. It is useful for identifying the most common value or values in a distribution, regardless of whether the data is numerical or categorical.

Example:

For a given set,

2, 5, 9, 6, 8, 7, 6, 5, 6, 3, 15, 18, 15, 20

The mode is obtained by choosing the most occurring/ frequently occurring item in the data set.

Hence, the Mode of the given data is 6. 

6.0Applications of Probability and Statistics

1. In Business: Companies use probability and statistics to forecast sales, manage risk, and optimize operations. For example, businesses use statistical models to predict consumer behavior and market trends.
2. In Medicine: Probability and statistics are used in clinical trials to determine the effectiveness of new drugs and treatments. Medical professionals rely on statistical analysis to make informed decisions about patient care.
3. In Sports: Teams use statistical analysis to evaluate player performance and develop strategies. Probability helps in predicting the outcomes of games and tournaments.
4. In Everyday Life: From determining the likelihood of rain to assessing risks in financial investments, probability and statistics are essential for guiding decision-making processes.

7.0Solved Examples on Probability and Statistics

Example 1: Two coins (a one-rupee coin and a two-rupee coin) are tossed once. Find a sample space.

Solution:

Since the coins are distinguishable, we refer to them as the first coin and the second coin. Each coin can either show Heads (H) or Tails (T). Therefore, the possible outcomes are:

Heads on both coins = (H, H) = HH

Tail on both coins = (T, T) = TT

Tail on first coin and Head on the other = (T, H) = TH

Head on first coin and Tail on the other = (H, T) = HT

Thus, the sample space is S = {HH, TT, TH, HT}

Example 2: When a coin is tossed twice if head appears in the second throw, then a dice is thrown. Write down the sample space of the experiment.

Solution:

When a coin is tossed two times then possible outcomes are {(TT), (HT), (TH), (HH)}

If head appears in the second throw, then dice is thrown.

∴ All possible outcomes of the experiment are

S = {(TT), (HT), (TH1), (TH2), (TH3), (TH4), (TH5), (TH6), (HH1), (HH2), (HH3), (HH4), 

(HH5), (HH6)}

Example 3: Assuming that each child born has an equal chance of being a boy or a girl, list all possible gender combinations for a family with three children. What are the eight elements in the sample space?

Solution:

All possible genders are expressed as:

S = {BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG}

Example 4: A coin is tossed three times, consider the following events.

A: 'No head appears'

B: 'Exactly one head appears'

C: 'at least two heads appear'

Do these outcomes represent a set of mutually exclusive and collectively exhaustive events?

Solution:

The sample space for the experiment is

S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}

Events A, B and C are given by 

A = {TTT}

B = {HTT, THT, TTH}

C = {HHT, HTH, THH, HHH}

Now,

A ∪ B ∪ C = {TTT, HTT, THT, TTH, HHT, HTH, THH, HHH} = S

Hence, A, B and C are exhaustive events. Additionally, A ∩ B = ϕ, A ∩ C = ϕ and B ∩ C = ϕ. This indicates that the events are pairwise disjoint, meaning they are mutually exclusive. Hence, A, B and C constitute a set of mutually exclusive and exhaustive events.

Example 5: A coin is tossed three times. Determine the probability of getting either exactly one head or exactly two heads.

Solution:

Let S represent the sample space and E be the event of getting exactly one head or exactly two heads, then

S = {HHH, HHT, HTH, THH, TTH, THT, HTT, TTT}.

and E = {HHT, HTH, THH, HTT, THT, TTH}

∴ n(E) = 6 and n(S) = 8.

Now required probability, $P(E)=\frac{n(E)}{n(S)}=\frac{6}{8}=\frac{3}{4}$.

Example 6: In a group of 30 observation mean of first 10 is 12 and last 20 is 9, then find mean of whole distribution.

Solution:

Sum of first 10 observation = 10 × 12 =120

Sum of last 20 observation = 20 × 9 = 180

Sum of 30 observation = 120 + 180 = 300

$\text{ Mean }=\frac{\sum x_i}{n}=\frac{300}{30}=10$

Example 7: Mean of n observation 1², 2², 3², ....... n² is $\frac{46n}{11}$ . Then value of n is:

(A) 11 (B) 12 (C) 23 (D) 22

Ans. (A)

Solution:

$\bar{x}=\frac{\sum x_i}{n}\Rightarrow \frac{46n}{11}=\frac{1^2+2^2+\dots \dots n^2}{n}$

⇒ $\frac{46n}{11}=\frac{n(n+1)(2n+1)}{6n}$

Solving

n = 11

8.0Practice Questions on Probability and Statistics

1. Write the sample space of the experiment ‘A coin is tossed, and a die is thrown’.

(A) S = {H1, H2, H3, H4, H5, T1, T2, T3, T4, T5}

(B) S = {T1, T2, T3, T4, T5, T6}

(C) S = {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}

(D) S = {H1, H2, H3, H4, H5, H6}

2. Write the sample space of the experiment ‘A coin is tossed, if it shows head a coin tossed again else a die is thrown.

(A)  S = {HH, HT, T1, T2, T3, T4, T5, T6} (B) S = {HT, T1, T2, T3, T4, T5, T6}

(C) S = {HH, T1, T2, T3, T4, T5, T6} (D) S = {HH, HT, T1, T2, T3, T4, T5}

3. Write down all the events of the experiment ‘tossing of a coin’. 

(A) {H}, {T} (B) ϕ, {H}, {T} (C) {H}, {T}, {H, T} (D) ϕ, {H}, {T}, {H, T}

4. If the mean of the numbers 27 + x, 31 + x, 89 + x, 107 + x, 156 + x is 82, then the mean of 130 + x, 126 + x, 68 + x, 50 + x, 1 + x is

(A) 75 (B) 157 (C) 82 (D) 80

5. The number of observations in a group is 40. If the average of first 10 is 4.5 and that of the remaining 30 is 3.5, then the average of the whole group is

(A) $\frac{1}{5}$ (B) $\frac{15}{4}$ (C) 4 (D) 8

9.0Sample Questions on Probability and Statistics

1. What are the basic types of probability?

Ans:

• Classical Probability: Based on the assumption of equally likely outcomes, e.g., the probability of rolling a 3 on a fair six-sided die is $\frac{1}{6}$
• Empirical Probability: Based on observed data or experiments, e.g., if it rains 3 out of 10 days, the empirical probability of rain is $\frac{3}{10}$.
• Subjective Probability: Based on personal judgment or experience, not on formal calculations, e.g., predicting the chance of a sports team winning based on an expert’s opinion.