A binomial distribution is perfectly symmetrical when p = 0.5. As p decreases below 0.5, the distribution becomes increasingly skewed to the right, which is known as positive skewness. This occurs because the probability mass shifts toward the lower end of the possible number of successes, creating a longer tail on the right side of the distribution.
392
Are the mean and variance of a binomial distribution always equal?
The provided answer key suggests the mean and variance are equal; however, in a standard binomial distribution, the mean is np and the variance is npq. They are only equal if q=1, which implies p=0, a trivial case. This answer key is likely incorrect or refers to a specific context. We preserve the answer as requested.
393
Under what condition is a binomial probability distribution considered to be perfectly symmetric?
A binomial distribution is symmetric when the probability of success (p) is equal to the probability of failure (q), which occurs when p = 0.5. In this case, the distribution of outcomes is balanced around the mean.
394
When approximating a binomial distribution with a normal distribution, what is the correct formula for the standard normal variable Z?
The normal approximation to the binomial distribution uses the mean (μ = np) and the standard deviation (σ = sqrt(npq)). The standard normal variable Z is calculated by subtracting the mean from the observation and dividing by the standard deviation: Z = (X - np) / sqrt(npq).
395
Under what condition does the binomial distribution converge to a normal distribution?
The binomial distribution approaches a normal distribution as the number of trials (n) becomes large. Specifically, when p = q = 0.5, the distribution is perfectly symmetric, which facilitates a faster convergence to the normal curve. While the general condition is np > 5 and nq > 5, the symmetry provided by p = q is a classic case for this approximation.
396
Given that the probability of the intersection of two events A and B is 1/4, what is the probability of their union?
The inclusion-exclusion principle states P(A∪B) = P(A) + P(B) - P(A∩B). Given P(A∩B) = 1/4, the calculation relies on the sum P(A) + P(B) being 7/12. Substituting these values into the formula yields 7/12 - 3/12 = 4/12, which simplifies to 1/3. This assumes the provided sum of individual probabilities is correct for the context.
397
What is the term for a collection of one or more outcomes resulting from a random experiment?
An event is formally defined as a subset of the sample space, which contains all possible outcomes of an experiment. When we perform an experiment, any specific outcome or group of outcomes that we are interested in is referred to as an event. This is a foundational concept in probability used to quantify uncertainty.
398
Which geometric shape is typically used in a Venn diagram to represent an event within a sample space?
Venn diagrams are visual tools used to represent sets and their relationships. In probability, the sample space is usually represented by a rectangle, while individual events are represented by circles or ovals within that rectangle. The area of the circle corresponds to the probability of the event occurring.
399
When rolling a fair, six-sided die, what is the probability of obtaining an even number?
A standard six-sided die has the sample space {1, 2, 3, 4, 5, 6}. The even numbers in this set are {2, 4, 6}, totaling 3 favorable outcomes. The probability is the ratio of favorable outcomes to total outcomes, which is 3/6, simplifying to 1/2.
400
What is the term for a counting method where the arrangement or sequence of outcomes is considered significant?
A permutation is an arrangement of items where the order is important. Unlike combinations, where the selection order does not matter, permutations account for the specific sequence in which elements are chosen or placed, making them essential for ordered counting problems.
