The conditional probability of event Y occurring given that event Z has already occurred is denoted as P(Y|Z). This notation signifies that the probability space is restricted to the occurrence of Z, and we are evaluating the likelihood of Y within that subset.
302
Given the conditional probability P(A|B) = 1/4 and the joint probability P(A∩B) = 1/5, what is the value of P(B)?
The definition of conditional probability is P(A|B) = P(A∩B) / P(B). Rearranging this formula to solve for P(B), we get P(B) = P(A∩B) / P(A|B). Substituting the given values: P(B) = (1/5) / (1/4). This is equivalent to (1/5) * (4/1), which equals 4/5. Thus, the probability of event B occurring is 0.8.
303
What is the expected value of a binomial distribution when the number of trials is equal to one?
In a binomial distribution, the expected value is defined as E(X) = np. When the number of trials (n) is equal to one, the formula simplifies to E(X) = 1 * p = p. Therefore, the expected value is simply the probability of success, which is the mean of a Bernoulli distribution, a special case of the binomial distribution where n=1.
304
What is the range of possible values that a binomial random variable can take, given n trials?
A binomial random variable represents the total number of successes in n independent Bernoulli trials. Since the number of successes cannot be negative and cannot exceed the total number of trials, the variable can take any integer value from 0 up to n inclusive.
305
Which condition regarding the probabilities p and q results in a positively skewed binomial distribution?
A binomial distribution is positively skewed when the probability of success p is less than 0.5, which implies p < q. In this scenario, the distribution has a tail extending toward the higher values, but the bulk of the probability is located at lower values.
306
How does the inequality of success probability (p) and failure probability (q) affect the skewness of a binomial distribution?
The skewness of a binomial distribution is given by (q-p)/sqrt(npq). If p < 0.5, the distribution is positively skewed (q > p). If p > 0.5, the distribution is negatively skewed (q < p). If p = 0.5, the distribution is symmetric. Thus, the skewness depends on the specific values of p and q.
307
Which probability distribution is characterized by having an equal mean and variance?
For a Poisson distribution with parameter λ, both the mean and the variance are equal to λ. This is a unique property of the Poisson distribution, which is often used to model the number of events occurring within a fixed interval of time or space. In contrast, for a binomial distribution, the variance is npq, which is always less than the mean np since q < 1.
308
How does the variance of a binomial distribution compare to its mean?
The mean of a binomial distribution is μ = np, and the variance is σ² = npq. Since q = 1 - p and p is a probability (0 < p < 1), it follows that q must also be between 0 and 1. Therefore, npq must be less than np, meaning the variance is always less than the mean for a binomial distribution.
309
Under what condition is a binomial distribution considered to be positively skewed?
A binomial distribution exhibits positive skewness when the probability of success (p) is less than the probability of failure (q). This results in a longer tail on the right side of the distribution, as the probability mass is concentrated toward the lower values of the random variable.
310
In the context of probability theory, how is the random variable in a negative binomial distribution typically classified?
The negative binomial distribution models the number of failures occurring before a specified number of successes in a sequence of independent Bernoulli trials. Because it counts the number of trials or failures, it is inherently a discrete random variable that represents the 'waiting time' until the r-th success occurs.
