Question

a 6 - sided die with faces labeled 1 to 6 will be rolled once. the 6 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 an odd number event b: rolling a number less than 5 event a and b: rolling an odd number and rolling a number less than 5 event a or b: rolling an odd number or rolling a number less than 5 (b) compute the following. (p(a)+p(b)-p(a\text{ and }b)=)

Explanation:

Step1: Identify outcomes for Event A

Odd - numbered outcomes are 1, 3, 5. So \(n(A)=3\). Total number of outcomes \(n(S) = 6\). Then \(P(A)=\frac{n(A)}{n(S)}=\frac{3}{6}\).

Step2: Identify outcomes for Event B

Numbers less than 5 are 1, 2, 3, 4. So \(n(B)=4\). Then \(P(B)=\frac{n(B)}{n(S)}=\frac{4}{6}\).

Step3: Identify outcomes for Event A and B

Odd - numbered and less than 5 are 1, 3. So \(n(A\cap B)=2\). Then \(P(A\cap B)=\frac{n(A\cap B)}{n(S)}=\frac{2}{6}\).

Step4: Identify outcomes for Event A or B

Using the formula \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\).

Step5: Calculate \(P(A)+P(B)-P(A\cap B)\)

Substitute the values: \(\frac{3}{6}+\frac{4}{6}-\frac{2}{6}=\frac{3 + 4-2}{6}=\frac{5}{6}\).

Answer:

Event A: Check boxes for 1, 3, 5; Probability \(\frac{3}{6}\)
Event B: Check boxes for 1, 2, 3, 4; Probability \(\frac{4}{6}\)
Event A and B: Check boxes for 1, 3; Probability \(\frac{2}{6}\)
Event A or B: Check boxes for 1, 2, 3, 4, 5; Probability \(\frac{5}{6}\)
(b) \(\frac{5}{6}\)