Q: Using the tree diagram for events A and B, which formula correctly expresses $$P(A \cap B) in $$terms of conditional probability?

$$P(A \cap B) = P(A) \times P(B \mid A)$$

Q: Given the binomial probability formula $$P(X = r) = \binom{n}{r} p^r q$$^{n-r}$$, what is the probability of exactly 2 successes in 5 independent trials with p = 0.3$?

$$\binom{5}{2} (0.3)^2$ (0.7)^3$ = 10 \times 0.09 \times 0.343 = 0.3087$$

Question 1

Refer to the Venn diagram with probabilities for events A and B (first image). What is the probability that a randomly selected individual is in either A or B?

Accepted Answer

$$0.56 + 0.15 + 0.03 = 0.74$$

Question 2

Based on the same Venn diagram, what is the probability that a randomly selected individual is in neither A nor B?

Accepted Answer

$$0.26$$

Question 3

Given the Venn diagram, what is $$P(A \mid B)$$, the probability that an individual is in A given they are in B?

Accepted Answer

$$\frac{0.15}{0.15 + 0.03} = \frac{0.15}{0.18} \approx 0.83$$

Question 4

In the Venn diagram, what is the probability that a randomly selected individual is in A but not in B?

Accepted Answer

$$0.56$$

Question 5

Suppose you roll two fair six-sided dice. What is the probability that the sum is 7? (Refer to the sample space grid shown in the images.)

Accepted Answer

$$\frac{6}{36} = \frac{1}{6}$$

Question 6

Using the tree diagram for events A and B, which formula correctly expresses $$P(A \cap B) in $$terms of conditional probability?

Accepted Answer

$$P(A \cap B) = P(A) 	imes P(B \mid A)$$

Question 7

Refer to the Pascal's Triangle image. What is the value of $$\binom{7}{3}$$?

Accepted Answer

$$35$$

Question 8

Given the binomial probability formula $$P(X = r) = \binom{n}{r} p^r q$$^{n-r}$$, what is the probability of exactly 2 successes in 5 independent trials with p = 0.3$?

Accepted Answer

$$\binom{5}{2} (0.3)^2$ (0.7)^3$ = 10 	imes 0.09 	imes 0.343 = 0.3087$$

Question 9

In the context of the scatterplot showing cost of women’s clothes vs. food costs, what type of association is most evident?

Accepted Answer

A positive linear association

Question 10

Given the scatterplot of max wind speed vs. central pressure, what is the likely sign of the correlation coefficient $$r$$?

Accepted Answer

Negative

Question 11

Which of the following best describes the difference between a bar chart and a histogram?

Accepted Answer

A bar chart is used for categorical data, while a histogram is used for quantitative data.

Question 12

In a boxplot, what does the interquartile range (IQR) represent?

Accepted Answer

The range of the middle 50% of the data

Question 13

If a distribution is right-skewed, which of the following is true?

Accepted Answer

The mean is greater than the median.

Question 14

Which sampling method involves dividing the population into subgroups and taking a simple random sample from each subgroup?

Accepted Answer

Stratified sampling

Question 15

Which of the following is NOT a requirement for a probability model?

Accepted Answer

All probabilities must be non-negative.

Question 16

Which of the following is a correct interpretation of a z-score of 2.5 for a student’s test score?

Accepted Answer

The student scored 2.5 standard deviations above the mean.

Comprehensive Study Notes for Statistics & Probability: Principles, Probability, Random Variables, Distributions, and Bivariate Analysis

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