Indicate the correct answer for the combination from the following regarding the conditions for the applicability of a binominal distribution: (a) There are n independent trials (b) Each trial has only two possible outcomes (c) The probabilities of two outcomes do not remain constant (d) The trials are independent Which of the following options is correct?
Understanding Binomial Distribution Conditions
The binomial distribution is a fundamental concept in probability theory and statistics. It is used to model the probability of obtaining a certain number of successes in a fixed number of independent trials, where each trial has only two possible outcomes.
For a scenario to follow a binomial distribution, several specific conditions must be met. Let's examine the conditions provided in the question:
- (a) There are n independent trials
- (b) Each trial has only two possible outcomes
- (c) The probabilities of two outcomes do not remain constant
- (d) The trials are independent
Analyzing Each Condition for Binomial Distribution
Let's analyze each condition to determine if it is required for a binomial distribution:
- (a) There are n independent trials: This is a crucial condition. The number of trials, denoted by 'n', must be fixed and known beforehand. The trials must also be independent, meaning the outcome of one trial does not affect the outcome of any other trial.
- (b) Each trial has only two possible outcomes: This is another essential condition. Each trial must result in one of only two mutually exclusive outcomes. These outcomes are typically labeled as "success" and "failure".
- (c) The probabilities of two outcomes do not remain constant: This condition is incorrect for a binomial distribution. For a binomial distribution to apply, the probability of success (denoted as 'p') must remain the same for every trial. Consequently, the probability of failure (denoted as '1-p') also remains constant across all trials.
- (d) The trials are independent: This condition reinforces point (a) and is absolutely necessary. Independence of trials ensures that the outcome of previous trials does not influence the outcome of subsequent trials.
Identifying the Correct Combination of Conditions
Based on the analysis, the necessary conditions for the applicability of a binomial distribution from the given list are:
- (a) There are n independent trials (Fixed number of independent trials)
- (b) Each trial has only two possible outcomes (Success/Failure)
- (d) The trials are independent (This is the same as part of condition (a), but listed separately)
Condition (c) is the opposite of a requirement for the binomial distribution.
Therefore, the correct combination of applicable conditions is (a), (b), and (d).
Summary of Binomial Distribution Conditions
| Condition | Description | Required for Binomial Distribution? |
|---|---|---|
| Fixed Number of Trials (n) | The experiment consists of a fixed number of trials. | Yes |
| Independent Trials | The outcome of each trial is independent of the outcomes of other trials. | Yes |
| Two Outcomes Per Trial | Each trial has only two possible, mutually exclusive outcomes (Success/Failure). | Yes |
| Constant Probability | The probability of success (p) is the same for each trial. | Yes |
Comparing this with the given options, option 3 which states (a), (b) and (d) represents the correct set of conditions from the list provided in the question.
Revision Table: Key Binomial Distribution Concepts
| Concept | Explanation |
|---|---|
| Binomial Experiment | An experiment satisfying the four key conditions: fixed trials, independent trials, two outcomes per trial, constant probability of success. |
| 'n' | The fixed number of trials. |
| 'p' | The constant probability of success on a single trial. |
| '1-p' (or 'q') | The constant probability of failure on a single trial. |
| Random Variable (X) | The number of successes in 'n' trials. |
| Probability Mass Function (PMF) | The formula used to calculate the probability of getting exactly k successes in n trials: \(P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}\) |
Additional Information: Applications of Binomial Distribution
The binomial distribution is widely used in various fields where experiments involve repeated independent trials with two outcomes and a constant probability of success. Some examples include:
- Quality Control: Checking if a certain number of items in a batch are defective (defective/not defective).
- Medicine: Testing the effectiveness of a drug (effective/not effective) on a number of patients.
- Surveys: Analyzing the responses to a yes/no question from a sample group.
- Genetics: Studying the inheritance of a specific trait (present/absent) in offspring.
- Sports: Analyzing a player's success rate (hit/miss, goal/no goal) over a series of attempts.
Understanding the conditions for the binomial distribution is crucial for correctly applying it to real-world problems and interpreting the results.
Important Questions from Binomial Distribution
In which of the following practical situations, Poisson Distribution can be used?
A. Number of customers arriving at the super markets per hour.
B. Number of typographical errors per page in a typed material.
C. Number of accidents taking place per day on a busy road.
D. Dice throwing problems.
E. Number of defective material say, blades, etc. in a packing manufactured by a good concern.
Choose the most appropriate answer from the options given below:
For Binomial distribution, n = 10 and p = 0.6, E(X 2) (second moment about origin) is:
The mean and variance of binomial distribution B (x, n, p) are 4 and \(\dfrac{4}{3}\) respectively. What is the probability of getting 2 successes?
Find out the fallacy if any in the statement:
“The mean and the variance of a binomial distribution is 16.2 and 29.4 respectively.”
If log e(x) is normally distributed with mean 1 and variance 4, then P(0.5 < x < 2) is:
(where the area between z = 0 and z = 0.25 is 0.0987 and the area between z = 0 and z = 0.5 is 0.1915)
