সম্ভাবনা MCQ Quiz in বাংলা - Objective Question with Answer for Probability - বিনামূল্যে ডাউনলোড করুন [PDF]

The correct answer is (1) Classical approach to probability

Key PointsProbability: When we don't know how an event will turn out, we can talk about the probabilities of certain outcomes—how likely they are. 

Statistics is the study of events that follow a probability distribution.

Additional Information

Four Approaches to probability:

1. Classical approach to probability: An approach appropriate for experiments that can be described with equally likely outcomes.
2. Statistical approach to probability: In this approach, probabilities of events were based on the results of actual experiments.
3. Subjective approach to probability: An approach appropriate to represent an individuals judgments based on facts combined with personal evaluations of other information.
4. Modern approach to probability: It includes both the classical and statistical definition as particular cases and overcomes the deficiencies of each of the approaches.

Therefore, The correct answer is (1) the Classical approach to probability

The correct answer is \(\frac{8}{16575}\)

Key Points

• The probability of drawing a king, a queen and a jack in order is the product of the probabilities of drawing each card. 
• There are 4 kings, 4 queens, and 4 jacks in a standard deck of 52 cards, so the probability of drawing any one of them is 4/52.
  • The first card can be any of the 3 cards in order. The probability of drawing a king first is \(\frac{4}{52}\). 
  • The second card can be either a queen or a jack, but it cannot be the same as the first card. 
• There are 4 queens or jacks left, and 51 cards in total, so the probability of drawing the second card is \(\frac{4}{51}\). 
• The same logic applies for the third card, and the probability of drawing a queen or a jack is \(\frac{4}{50}\).
  • Therefore, the probability of drawing a king, a queen, and a jack in order is \(\frac{4}{52} * \frac{4}{51} *\frac{4}{50} = \frac{8}{16575}\)}

Additional Information

• The number of each card in the deck affects the probability of drawing them.
• The order of drawing the cards affects the final probability.
• This problem assumes that the cards are drawn without replacement, which means that once a card is drawn, it is not returned to the deck.
• The problem calculates the probability of drawing the cards in a specific order, but there are other ways to calculate the probability of drawing a specific combination of cards, regardless of order.

Explanation:

Assumptions in the game theory:

• Payoffs are known and fixed

All players behave rationally and intelligently.

• The understand and seek to maximize gains and minimize losses.
• they are flawless in calculating which actions will maximize their payoffs.

The rules of game are common knowledge:

• Each player knows the set of players, strategies and payoffs from all possible combinations of strategies. Call this information X.
• Common knowledge means that each player knows that all players know X.

Concept:

This problem involves a queuing system with Poisson arrivals and exponential service times (M/M/1 queue). The probability that an arrival does not have to wait for service is equal to the probability that the system is idle (i.e., no customers are being served).

Given:

• Average time between arrivals (\(\lambda^{-1}\)) = 12 min → Arrival rate, \(\lambda = \frac{1}{12}\) customers per minute
• Average service time (\(\mu^{-1}\)) = 5 min → Service rate, \(\mu = \frac{1}{5}\) customers per minute

Step 1: Calculate the Traffic Intensity (ρ)

The traffic intensity is the ratio of arrival rate to service rate:

\[ \rho = \frac{\lambda}{\mu} = \frac{\frac{1}{12}}{\frac{1}{5}} = \frac{5}{12} ≈ 0.4167 \]

Step 2: Determine the Probability of System Being Idle (P₀)

For an M/M/1 queue, the probability that the system is idle (no customers in the system) is:

\[ P_0 = 1 - \rho = 1 - \frac{5}{12} = \frac{7}{12} ≈ 0.5833 \]

The probability that an arrival does not have to wait for service is equal to the probability that the system is idle, which is P₀ ≈ 0.5833

Explanation:

Given the data:

• Arrival rate (λ) = 120 customers per hour
• Service rate (μ) = 150 customers per hour

The utilization factor (ρ) is the ratio of the arrival rate to the service rate. It represents the proportion of time the server is busy. The formula for the utilization factor is:

ρ = λ / μ

Substituting the given values:

ρ = 120 / 150

Simplifying this fraction:

ρ = 0.8

The utilization factor (ρ) indicates that the server is busy 80% of the time. Therefore, the proportion of time the server is idle (1 - ρ) can be calculated as:

Idle Time Proportion = 1 - ρ

Substituting the value of ρ:

Idle Time Proportion = 1 - 0.8

Idle Time Proportion = 0.2

Therefore, the proportion of time the server is idle is 0.2

The correct answer is 0.736

Important Points

Total number of misprints=100

Total number of pages in a book = 100

Expected mean value=

Poisson distribution formula :

Using the formula

Hence, the probability that a page selected at random will contain at most one misprint will be 0.736.

Let us assume - 

• The event of selecting a red ball is denoted by 'R'.
• The event of selecting the bag A is denoted by 'L'.
• The event of selecting the bag B is denoted by 'M'.

We know that - 

• Bag-L has 2 white and 3 red balls
• Bag-M has 4 white and 5 red balls.

Hence, 

• Probability of selecting Bag L and Bag M = \(P(L) = P(M) = \frac{1}{2}\)
• Probability of selecting Red ball from Bag L = \(P(\frac{R}{L}) = \frac{3}{5}\)
• Probability of selecting Red ball from Bag M = \(P(\frac{R}{M}) = \frac{5}{9}\)

Applying Baye's Theorem, we get = \(P(\frac{B}{R}) = \frac{P(\frac{R}{B}) \times P(B)}{P(R)} = \frac{P(\frac{R}{B}) \times P(B)}{P(\frac{R}{A}) \times P(A) + P(\frac{R}{B}) \times P(B)}\)

=> \(\frac{\frac{5}{9} \times \frac{1}{2}}{\frac{3}{5} \times \frac{1}{2} + \frac{5}{9} \times \frac{1}{2}}\)

=> \(\frac{25}{52}\)

The probability that at least one of the students P, Q, and R solves the problem is the complement of the probability that none of them solves it.

The probability that P does not solve the problem is 1 - 1/3 = 2/3
The probability that Q does not solve the problem is 1 - 1/4 = 3/4
The probability that R does not solve the problem is 1 - 1/5 = 4/5

So, the probability that none of the students solve the problem is the product of these probabilities:
\(P(\text{None Solve}) = \frac{2}{3} \times \frac{3}{4} \times \frac{4}{5} = \frac{24}{60} = \frac{2}{5}\)

Therefore, the probability that at least one of the students solves the problem is:
\(P(\text{At Least One Solves}) = 1 - P(\text{None Solve}) = 1 - \frac{2}{5} = \frac{3}{5}\)

Hence, the probability that the problem will be solved by at least one of the students is \(\frac{3}{5}\) = 0.6.

Explanation:

Arrival rate (x) = 5/hour

Service rate \({\rm{\;}}\left( \mu \right) = \frac{{60}}{5} = 12/hour\)

The arriving customer will get a seat when there is maximum one person waiting i.e., when there is no customer, or one in system or two in system (one served and one waiting).

Thus,

Required probability = P₀ + P₁ + P₂

Required probability \(= \left( {1 - \frac{\lambda }{\mu }} \right) + \left( {\frac{\lambda }{\mu }} \right)\left( {1 - \frac{\lambda }{\mu }} \right) + {\left( {\frac{\lambda }{\mu }} \right)^2}\left( {1 - \frac{\lambda }{\mu }} \right)\)

Required probability \( = \left( {1 - \frac{5}{{12}}} \right) + \left( {\frac{5}{{12}}} \right)\left( {1 - \frac{5}{{12}}} \right) + {\left( {\frac{5}{{12}}} \right)^2}\left( {1 - \frac{5}{{12}}} \right)\)

Required probability \(= \frac{7}{{12}} + \frac{5}{{12}} \times \frac{7}{{12}} + {\left( {\frac{5}{{12}}} \right)^2} \times \frac{7}{{12}}\)

∴ Required probability = 0.927

Concept:

μ = service rate

P(more than T) = e^-μT

Calculation:

μ = 18 customers/hour

T = 12 minutes = 0.2 hour

∴ P (more than 12 minutes) = e^{-18 × 0.2}

∴ P = 2.73 %