৪৯তম বিসিএস ⎯ পরিসংখ্যান [৯৮১]

Sample space = {1, 2, 3, 4, 5, 6} → total outcomes = 6.

Even outcomes = {2, 4, 6} → favorable outcomes = 3. 

So, P(E) = 3/6
               = 1/2

Two events are mutually exclusive if they cannot happen together.

Only option (a) satisfies this (a coin cannot be both Head and Tail at the same time).

By complement rule:

P(Not Rain) = 1−P(Rain)
                     = 1−0.7 
                     = 0.3

In case of Tossing a coin: P(Head) = 1/2

In case of Rolling a die: P(6) = 1/6

এখানে, দুটা ঘটনা স্বাধীন।
অর্থাৎ, একটার ঘটার সাথে আরেকটার ঘটার সম্ভাবনা নির্ভর করে না।

Independent event এর ক্ষেত্রে আমরা জানি, P(A∩B)= P(A)×P(B)

So, Since independent → Multiply: P(Head∩6) 
= (1/2)×(1/6) 
= 1/12

There are 2 blue balls (favorable outcomes) and a total of 5 balls (3 red + 2 blue) in the bag.

The probability is the number of favorable outcomes divided by the total number of outcomes, which is 2/5. 

A fair coin has two equally likely outcomes: heads or tails.

But they do not contain "1". "1" is an outcome of a dice.

So, it is an impossible event. 

P(impossible event) = 0.

The numbers greater than 4 on a seven sided die are 5, 6, 7. 

So, there are 3 favorable outcomes.

The total number of possible outcomes is 7.

The probability is 3/7.

Independence means occurrence of one event does not affect the other, so joint probability is the product of individual probabilities.

Source: Business Statistics, Md. Abdul Aziz.

The probability of the sample space S (sure event) is always 1.

মানে নমুনাক্ষেত্র বা স্যাম্পল স্পেসের যেকোন পয়েন্ট ঘটবেই তার সম্ভাবনা ১.

Source: Live mcq class lecture 

প্রথম অপশনে less than 1, greater than 0 বলা আছে।

সঠিক হবে- less than or equal 1, greater than or equal 0.

শেষ অপশনে অর্ধেক শর্ত লেখা আছে, greater equal 0- এটুকুও লিখতে হবে।

তাই শুধু b সঠিক। 

For mutually exclusive events: P(A∪B) = P(A)+P(B)
= 0.4+0.5 
= 0.9

A certain event always occurs, so its probability is 1.

P(Tea∪Coffee)

= P(Tea) + P(Coffee) – P(Tea ∩ Coffee)
= 0.70 + 0.50 – 0.20 
= 1.00

Independence definition = multiplication rule.

For two independent event->

P(A∩B) = P(A) × P(B)

If first die = 6, second die can be 1–6.

Possible sums = {7, 8, 9, 10, 11, 12}.

Favorable sums ≥ 10 → {10, 11, 12} → 3 outcomes.

Total possible outcomes = 6.

So, probability = 3/6 = 1/2 

The classical approach assumes all outcomes are equally likely.

The empirical approach defines probability as the relative frequency of an event in a large number of trials.

Source: Business Statistics (MP Gupta, SP Gupta.)  

Probability is always non-negative; this is not a valid axiom.

Bayes’ theorem is used to revise prior beliefs (prior probabilities) into posterior probabilities when new evidence is observed.

Conditional probability is the probability of an event A under the condition that another event B has already happened.

-Marginal probability = probability of a single event.

-Conditional probability = probability of an event given another.

-Joint probability = probability of both (or more) occurring simultaneously.

Joint and conditional both looks at two events.

Joint probability is about both events occurring; conditional is about one event given the other.

Source: Business Statistics (MK Roy.) 

a) It represents A union B. 

b) This is Marginal probability. 

c) correct definition of joint probability 

d)  It represents conditional probability 

From the formula of Bayes theorem we can see, it allows us to reverse conditions. 

Classical probability assumes equally likely outcomes (e.g., rolling a fair die).

Frequentist probability defines probability as the long-run relative frequency of events after repeated trials.

Explanation:
Mutually exclusive → no overlap P(A∩B)=0
Independent → no influence on probability
P(A∣B)=P(A).
So, both are not same.

Conditional independence means:

Once we know C, learning about B gives no extra information about A.
Mathematically:
P(A∣B,C)=P(A∣C) (That’s option c).

Sometimes, two events look dependent at first, but if we bring in another event C, they can become independent. That's option B

Suppose A and B are mutually exclusive: Then, P(A∩B)=0

But if they were also independent, then:

P(A∩B)=P(A)⋅P(B) 
so, একই সাথে mutually exclusive ও independent হলে,
  P(A)⋅P(B)= 0 হতে হবে। কিন্তু, 
If both P(A), P(B) are greater than 0, then their product is greater than 0, which contradicts mutual exclusivity.
That means at least one of them must be 0. অর্থাৎ সহজ কথায়, দুটা অশূন্য সম্ভাবনা বিশিষ্ট ঘটনা একই সাথে mutually exclusive ও independent হতে পারে না। 

Explanation:
Law of total probability states:
P(A)=∑P(A∣Bi)P(Bi), where {Bi​} partition the sample space.

Explanation:

Without replacement, the second draw depends on the first draw, making events dependent.

Explanation:

By definition, for independent events: P(A∩B) = P(A)P(B). Only this is true always.
a সঠিক হতো যদি লিখতাম-
P(A|B)= P(A)

Explanation:

Marginal probability = total probability of one variable by summing (or integrating) over the other variable(s) in a joint distribution.

Explanation:
Subjective probability quantifies personal belief or expert judgment (e.g., “60% chance it will rain tomorrow” without empirical trials)
Source: Business Statistics, SP Gupta, MP Gupta.

Explanation:
Simple = single outcome (e.g., “rolling a 3”).
Compound = multiple outcomes (e.g., “rolling an odd number”).

Explanation:

Complementary events are mutually exclusive and exhaustive, so:

P(A) + P(Aˉ) = 1

Explanation:

Independence means occurrence of one does not change probability of the other:

If, P(A∣B) = P(A)

Explanation:

Impossible event has probability 0 (die cannot show 7).

Explanation:

Equally likely means each outcome has the same chance.

Explanation:
Exhaustive events = together they cover all possible outcomes.

Explanation:

Pairwise mutually exclusive means any two events cannot occur together. But it does not guarantee that they are collectively exhaustive or independent.

Explanation:

Collectively exhaustive = at least one of the events must happen (their union = sample space).
So, If A and B are exhaustive event, then AUB= Total Sample space 

Explanation:
Complementary events are mutually exclusive and exhaustive, but not independent (since knowing A happened makes P(Aˉ)=0)

Explanation:

Marginal probability = probability of a single variable (ignores other dimensions).
Source: Lecture slide math

Explanation:

Bayes’ theorem denominator = P(B). This is usually computed using law of total probability:

P(B) = ∑P(B∣Ai)P(Ai)

Explanation:

Bayes’ theorem does not require events to be mutually exclusive. It just needs conditional and marginal probabilities.

Explanation:

Even = {2,4,6}, Odd = {1,3,5}. Together they cover all outcomes → exhaustive.

পূর্ণ ঘটনা” বা “সর্বসম্ভাব্য ঘটনা”

ব্যাখ্যা:

যখন একটি সেট ইভেন্ট (event) একত্রে সমস্ত সম্ভাব্য ফলাফলকে আচ্ছাদিত করে, তখন সেই ইভেন্টগুলোকে পূর্ণ (exhaustive) ইভেন্ট বলা হয়।
সহজভাবে বলতে গেলে: কমপক্ষে একটি ইভেন্ট ঘটবেই, কারণ তারা সব সম্ভাব্য ফলাফলকে কভার করে।
উদাহরণ:

একটি পাশা নিক্ষেপ করলে:

Event A: ফলাফল জোড় সংখ্যা (2,4,6)
Event B: ফলাফল বিজোড় সংখ্যা (1,3,5)
→ A∪B = {1,2,3,4,5,6} = সব সম্ভাব্য ফলাফল → পূর্ণ ঘটনা।

Explanation:
This is a reverse probability problem:
we want 
P(Defective | Test say defective).
Bayes’ theorem updates prior probability given new evidence.

Explanation:

Double = {1,1}, {2,2}, …, {6,6} → 6 outcomes.
Not double = remaining 36−6=30 outcomes.
Together they cover all 36 outcomes → exhaustive → probability = 1.

Explanation:

P(D)=0.02,P(Dˉ)=0.98
P(+∣D)=0.95,  P(+∣Dˉ)=0.05
Using Bayes:

P(D∣+)=(0.95×0.02)/((0.95×0.02)+(0.05×0.98)) = 0.019/​0.068≈0.28
so, 28% chance of actually having the disease.

Explanation:

P(Head)=1/2,  P(6)=1/6
Since independent:

P(Head and 6)=1/2​×1/6​=1/12​