Asked by John Doe
What is a binomial random variable and how is it calculated?
Answers
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GPT-4o mini
A **binomial random variable** is a type of discrete random variable that arises from a binomial experiment. A binomial experiment has the following characteristics:
1. **Fixed number of trials (n)**: The experiment is conducted a predetermined number of times.
2. **Two possible outcomes**: Each trial results in one of two outcomes, often labeled as "success" and "failure."
3. **Constant probability (p)**: The probability of success (denoted as \(p\)) is the same for each trial.
4. **Independent trials**: The outcomes of the trials are independent of one another.
The binomial random variable \(X\) represents the number of successes in these \(n\) trials.
### Probability Mass Function (PMF)
The probability of observing exactly \(k\) successes (where \(k\) can range from 0 to \(n\)) in \(n\) trials is given by the **binomial probability formula**:
\[
P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}
\]
Where:
- \(P(X = k)\) is the probability of getting exactly \(k\) successes.
- \(n\) is the total number of trials.
- \(k\) is the number of successes (where \(0 \leq k \leq n\)).
- \(p\) is the probability of success on an individual trial.
- \((1 - p)\) is the probability of failure.
- \(\binom{n}{k}\) is a binomial coefficient calculated as:
\[
\binom{n}{k} = \frac{n!}{k! (n - k)!}
\]
### Example Calculation
Suppose you want to find the probability of getting exactly 3 heads in 10 flips of a fair coin, where the probability of heads (success) is \(p = 0.5\):
1. Set \(n = 10\) (the number of trials).
2. Set \(k = 3\) (the desired number of successes).
3. Calculate \(P(X = 3)\):
\[
P(X = 3) = \binom{10}{3} (0.5)^3 (0.5)^{10 - 3}
\]
Calculating the binomial coefficient:
\[
\binom{10}{3} = \frac{10!}{3!(10 - 3)!} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120
\]
Putting it all together:
\[
P(X = 3) = 120 \cdot (0.5)^3 \cdot (0.5)^7 = 120 \cdot (0.5)^{10} = 120 \cdot \frac{1}{1024} = \frac{120}{1024} = \frac{15}{128} \approx 0.1172
\]
Thus, the probability of getting exactly 3 heads in 10 flips of a fair coin is approximately \(0.1172\).
1. **Fixed number of trials (n)**: The experiment is conducted a predetermined number of times.
2. **Two possible outcomes**: Each trial results in one of two outcomes, often labeled as "success" and "failure."
3. **Constant probability (p)**: The probability of success (denoted as \(p\)) is the same for each trial.
4. **Independent trials**: The outcomes of the trials are independent of one another.
The binomial random variable \(X\) represents the number of successes in these \(n\) trials.
### Probability Mass Function (PMF)
The probability of observing exactly \(k\) successes (where \(k\) can range from 0 to \(n\)) in \(n\) trials is given by the **binomial probability formula**:
\[
P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}
\]
Where:
- \(P(X = k)\) is the probability of getting exactly \(k\) successes.
- \(n\) is the total number of trials.
- \(k\) is the number of successes (where \(0 \leq k \leq n\)).
- \(p\) is the probability of success on an individual trial.
- \((1 - p)\) is the probability of failure.
- \(\binom{n}{k}\) is a binomial coefficient calculated as:
\[
\binom{n}{k} = \frac{n!}{k! (n - k)!}
\]
### Example Calculation
Suppose you want to find the probability of getting exactly 3 heads in 10 flips of a fair coin, where the probability of heads (success) is \(p = 0.5\):
1. Set \(n = 10\) (the number of trials).
2. Set \(k = 3\) (the desired number of successes).
3. Calculate \(P(X = 3)\):
\[
P(X = 3) = \binom{10}{3} (0.5)^3 (0.5)^{10 - 3}
\]
Calculating the binomial coefficient:
\[
\binom{10}{3} = \frac{10!}{3!(10 - 3)!} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120
\]
Putting it all together:
\[
P(X = 3) = 120 \cdot (0.5)^3 \cdot (0.5)^7 = 120 \cdot (0.5)^{10} = 120 \cdot \frac{1}{1024} = \frac{120}{1024} = \frac{15}{128} \approx 0.1172
\]
Thus, the probability of getting exactly 3 heads in 10 flips of a fair coin is approximately \(0.1172\).
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