Question

an 8 - sided die with faces labeled 1 to 8 will be rolled once. the 8 possible outcomes are listed below. note that each outcome has the same probability. complete parts (a) through (c). write the probabilities as fractions. (a) check the outcomes for each event below. then, enter the probability of the event. event a: rolling a number greater than 2 event b: rolling an even number event a or b: rolling a number greater than 2 or rolling an even number event a and b: rolling a number greater than 2 and rolling an even number (b) compute the following. p(a)+p(b)-p(a and b)=□ (c) select the answer that makes the equation true. p(a)+p(b)-p(a and b)= select

Explanation:

Step1: Find P(A)

Event A is rolling a number greater than 2. The favorable outcomes are 3, 4, 5, 6, 7, 8. So the number of favorable outcomes $n(A)=6$. The total number of outcomes $n = 8$. Then $P(A)=\frac{n(A)}{n}=\frac{6}{8}=\frac{3}{4}$.

Step2: Find P(B)

Event B is rolling an even - number. The favorable outcomes are 2, 4, 6, 8. So the number of favorable outcomes $n(B)=4$. Then $P(B)=\frac{n(B)}{n}=\frac{4}{8}=\frac{1}{2}$.

Step3: Find P(A and B)

Event A and B is rolling a number greater than 2 and even. The favorable outcomes are 4, 6, 8. So the number of favorable outcomes $n(A\cap B)=3$. Then $P(A\cap B)=\frac{n(A\cap B)}{n}=\frac{3}{8}$.

Step4: Compute P(A)+P(B) - P(A and B)

Substitute the values of $P(A)$, $P(B)$ and $P(A\cap B)$ into the formula: $P(A)+P(B)-P(A\cap B)=\frac{3}{4}+\frac{1}{2}-\frac{3}{8}=\frac{6 + 4-3}{8}=\frac{7}{8}$.

Step5: Analyze the relationship for part (c)

By the addition rule of probability, $P(A\cup B)=P(A)+P(B)-P(A\cap B)$. And $P(A\cup B)$ (rolling a number greater than 2 or an even number) has 7 favorable outcomes (3, 4, 5, 6, 7, 8, 2), so $P(A\cup B)=\frac{7}{8}$.

Answer:

(a)
Event A: Check 3, 4, 5, 6, 7, 8; Probability $\frac{3}{4}$
Event B: Check 2, 4, 6, 8; Probability $\frac{1}{2}$
Event A or B: Check 2, 3, 4, 5, 6, 7, 8; Probability $\frac{7}{8}$
Event A and B: Check 4, 6, 8; Probability $\frac{3}{8}$
(b) $\frac{7}{8}$
(c) $P(A\cup B)$ (the probability of event A or B)