8 Questions
What is the probability mass function (PMF) of the binomial distribution?
The PMF of the binomial distribution is given by: $P(X=x) = \binom{n}{x} p^x q^{(n-x)}$, for $x = 0, 1, 2, \ldots, n$, and 0 otherwise.
What is the mean of the binomial distribution?
The mean of the binomial distribution is $np$.
What is the variance of the binomial distribution?
The variance of the binomial distribution is $npq$, where $q = 1 - p$.
In the same example, what is the probability that Pat gets all 10 answers correct by guessing?
The probability that Pat gets all 10 answers correct is 0.0000001024.
In the example with the quiz consisting of 10 multiple-choice questions, what is the probability that Pat gets exactly 1 answer correct by guessing?
The probability that Pat gets exactly 1 answer correct is 0.2684.
In the example with the basket containing 20 good oranges and 80 bad oranges, if 3 oranges are drawn at random, what is the probability of getting exactly 2 good oranges?
The probability of getting exactly 2 good oranges is $\binom{3}{2} \left(\frac{20}{100}\right)^2 \left(\frac{80}{100}\right)^{(3-2)} = 0.2048$.
Suppose a random variable X follows a binomial distribution with parameters n and p. Express the mean of X in terms of its variance.
The mean of X can be expressed as $np = \frac{npq}{q}$, where the variance of X is npq.
In the example with the random sample of 5 students from a college where 20% are girls, what is the probability of having at most 2 girls in the sample?
The probability of having at most 2 girls in the sample is 0.94208.
Study Notes
Bernoulli Distribution
- The probability density function (P.D.F) of Binomial Distribution is: P(X=x) = 𝑛𝑐𝑥 𝑃𝑥 𝑞(𝑛−𝑥) = 0 ,otherwise
Properties of Binomial Distribution
- Mean of Binomial Distribution (B.D) = np
- Variance of B.D = npq
Example 1: Quiz with 10 MCQ
- Probability of Pat getting 1 answer correct: P(X=1) = 10𝑐 1 (0.2)1 (0.8)(10−1) = 0.2684
- Probability of Pat getting all 10 answers correct: P(X=10) = 10𝑐 10 (0.2)10 (0.8)(10−10) = 0.0000001024
Example 2: Random Sample of Students
- Probability of at most two girls in a random sample of 5 students: P(atmost two girls) = P(X=0) + P(X=1) + P(X=2) = 0.94208
Example 3: Drawing Oranges from a Basket
- Probability of drawing 3 oranges (good or bad) from a basket with 20 good and 80 bad oranges
Learn about the properties of Bernoulli distribution and Binomial distribution, including the probability mass function, mean, and variance. Solve a probability problem related to guessing answers in a quiz with multiple-choice questions.
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