Which of the following is the probability of \( x \) successes in a binomial distribution with number of trials \( n \) and probability of success as \( \theta \) ( \( 0 < \theta < 1 \) ) in each trial?
\( ^n c_x \theta^x (1 - \theta)^{n-x}, x = 0,1,2,\dots,n \)
Understanding Binomial Probability Distribution
The question asks for the probability of observing exactly \( x \) successes in a binomial distribution. A binomial distribution models the number of successes in a fixed number of independent trials, where each trial has only two possible outcomes: success or failure. These trials are often called Bernoulli trials.
Key Parameters of Binomial Distribution
- \( n \): The total number of independent trials.
- \( \theta \) (or \( p \)): The probability of success in a single trial. This probability remains constant for all trials, and \( 0 < \theta < 1 \).
- \( 1 - \theta \) (or \( q \)): The probability of failure in a single trial.
- \( x \): The number of successes we are interested in observing, where \( x \) can be any integer from 0 up to \( n \).
Binomial Probability Mass Function (PMF)
The formula that gives the probability of obtaining exactly \( x \) successes in \( n \) trials for a binomial distribution is known as the Probability Mass Function (PMF). This formula combines the number of ways to get \( x \) successes in \( n \) trials with the probability of getting exactly \( x \) successes and \( n-x \) failures in any specific order.
The formula is:
\( P(X = x) = <sup>n</sup> c_x \theta^x (1 - \theta)^{n-x} \)
Where:
- \( <sup>n</sup> c_x \) represents the number of combinations of choosing \( x \) successes from \( n \) trials. It is calculated as \( \frac{n!}{x!(n-x)!} \).
- \( \theta^x \) represents the probability of getting exactly \( x \) successes. Since the trials are independent, we multiply the probability of success (\( \theta \)) \( x \) times.
- \( (1 - \theta)^{n-x} \) represents the probability of getting exactly \( n-x \) failures. Since the trials are independent, we multiply the probability of failure (\( 1 - \theta \)) \( n-x \) times.
The possible values for \( x \) are \( 0, 1, 2, \dots, n \).
Evaluating the Given Options
Let's compare the standard binomial PMF formula with the given options:
- \( <sup>n</sup> p_x \theta^x (1 - \theta)^{n-x}, x = 0,1,2,\dots,n \)
- \( <sup>n</sup> c_x \theta^x (1 - \theta)^{n-x}, x = 0,1,2,\dots,n \)
- \( <sup>n</sup> c_x \theta (1 - \theta) \)
- \( <sup>n</sup> c_x \theta^x (1 - \theta)^x, x = 0,1,2,\dots,n \)
Analyzing each option:
- Option 1 uses \( <sup>n</sup> p_x \), which represents permutations (order matters). For a binomial distribution, we only care about the total number of successes, not the order in which they occur. Therefore, combinations \( <sup>n</sup> c_x \) should be used, not permutations. This option is incorrect.
- Option 2 uses \( <sup>n</sup> c_x \) for the combinations, \( \theta^x \) for the probability of \( x \) successes, and \( (1 - \theta)^{n-x} \) for the probability of \( n-x \) failures, and specifies the correct range for \( x \). This matches the standard formula for the binomial probability mass function. This option is correct.
- Option 3 uses \( <sup>n</sup> c_x \) but has incorrect exponents for \( \theta \) and \( (1 - \theta) \). It uses \( \theta^1 \) and \( (1 - \theta)^1 \) implicitly, which is not the general formula for \( x \) successes and \( n-x \) failures. This option is incorrect.
- Option 4 uses \( <sup>n</sup> c_x \) and \( \theta^x \), but the exponent for the probability of failure is \( x \) instead of \( n-x \). This is incorrect as there are \( n-x \) failures when there are \( x \) successes in \( n \) trials. This option is incorrect.
Based on the analysis, the correct formula for the probability of \( x \) successes in a binomial distribution with \( n \) trials and probability of success \( \theta \) is given by Option 2.
| Component | Meaning | Formula Term |
|---|---|---|
| Combinations | Number of ways to choose \( x \) successes from \( n \) trials | \( <sup>n</sup> c_x \) |
| Success Probability Product | Probability of getting \( x \) successes (assuming a specific order) | \( \theta^x \) |
| Failure Probability Product | Probability of getting \( n-x \) failures (assuming a specific order) | \( (1 - \theta)^{n-x} \) |
| Full Probability | Probability of getting exactly \( x \) successes in \( n \) trials (any order) | \( <sup>n</sup> c_x \theta^x (1 - \theta)^{n-x} \) |
Revision Table: Binomial Distribution Formula
It is crucial to remember the correct formula for binomial probabilities.
| Distribution | Parameters | Probability of x Successes | Possible values of x |
|---|---|---|---|
| Binomial | \( n \), \( \theta \) | \( P(X=x) = <sup>n</sup> c_x \theta^x (1 - \theta)^{n-x} \) | \( x = 0, 1, 2, \dots, n \) |
Additional Information: Binomial Distribution Properties
Beyond the probability formula, understanding other aspects of the binomial distribution is helpful for statistics and probability problems.
- Mean (Expected Value): The average number of successes expected in \( n \) trials is given by \( E(X) = n\theta \).
- Variance: The spread of the distribution is measured by the variance, which is \( Var(X) = n\theta(1 - \theta) \).
- Standard Deviation: The square root of the variance, \( SD(X) = \sqrt{n\theta(1 - \theta)} \).
- Assumptions: For a variable to follow a binomial distribution, the following conditions must be met:
- Fixed number of trials (\( n \)).
- Each trial has only two outcomes (success or failure).
- The probability of success (\( \theta \)) is constant for every trial.
- The trials are independent of each other.
Recognizing and correctly applying the binomial probability formula is fundamental when dealing with scenarios involving a fixed number of independent trials with two outcomes.
Similar Questions
Let \( X \) be a random variable whose probability distribution is given by the table:
X 1 3 5 7 P(X) \( \frac{1}{3} \) \( \frac{1}{6} \) \( \frac{1}{6} \) \( \frac{1}{3} \) Then variance of \( X \) is:
For a Binomial distribution \( B(n, p) \), \( \frac{E(x)}{V(x)} \) is equal to:
The probability that the problem is solved is:
The probability that either only Kalyani or Kashvi or Sara solves it is:
Important Questions from Probability
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B2D, C3F, E5J, G7N, ?, M13Z
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From the given options, at what angle are the hands of a clock inclined at 10 minutes to 2 (Smaller angle)?
In the given analogy, choose the number which will replace the question mark (?).
WSH : 5 : : KMJ : ?
