Question

1. you spin a spinner with sections labeled 1 - 15. find the probability of the event: even number.

a) \\( \frac { 1 } { 2 } \\) b) \\( \frac { 8 } { 15 } \\)
c) \\( \frac { 1 } { 3 } \\) d) \\( \frac { 7 } { 15 } \\)

1. you spin a spinner with sections labeled 1 - 15. find the probability of the event: 3 or 4.

a) \\( \frac { 1 } { 15 } \\) b) \\( \frac { 2 } { 15 } \\)
c) \\( \frac { 1 } { 5 } \\) d) \\( \frac { 4 } { 15 } \\)

1. you spin a spinner with sections labeled 1 - 15. find the probability of the event: even number or multiple of 3.

a) \\( \frac { 3 } { 5 } \\) b) \\( \frac { 7 } { 15 } \\)
c) \\( \frac { 1 } { 2 } \\) d) \\( \frac { 2 } { 3 } \\)

1. a bag contains 4 blue marbles, 5 yellow marbles, and 3 red marbles. if you draw one marble at random, find the probability of red.

a) \\( \frac { 1 } { 3 } \\) b) \\( \frac { 3 } { 5 } \\)
c) \\( \frac { 5 } { 12 } \\) d) \\( \frac { 1 } { 4 } \\)

Explanation:

Step1: Determine total number of outcomes

For spinner labeled \(1 - 15\), total number of outcomes \(n(S)=15\). For marble problem, total number of marbles \(n = 4 + 5+3=12\).

Step2: Analyze each problem

Problem 1: Even number

Even numbers from \(1 - 15\) are \(2,4,6,8,10,12,14\), so \(n(A)=7\). Probability \(P(A)=\frac{n(A)}{n(S)}=\frac{7}{15}\).

Problem 2: \(3\) or \(4\)

Let \(B\) be the event. \(n(B) = 2\) (i.e., \(3\) and \(4\)). Probability \(P(B)=\frac{n(B)}{n(S)}=\frac{2}{15}\).

Problem 3: Even number or multiple of \(3\)

Even numbers \(E=\{2,4,6,8,10,12,14\}\), \(n(E) = 7\). Multiples of \(3\) \(M=\{3,6,9,12,15\}\), \(n(M)=5\). \(E\cap M=\{6,12\}\), \(n(E\cap M) = 2\). By formula \(P(E\cup M)=P(E)+P(M)-P(E\cap M)=\frac{7}{15}+\frac{5}{15}-\frac{2}{15}=\frac{10}{15}=\frac{2}{3}\).

Problem 4: Red marble

Let \(R\) be the event. \(n(R) = 3\). Probability \(P(R)=\frac{n(R)}{n}=\frac{3}{12}=\frac{1}{4}\).

Answer:

1. d) \(\frac{7}{15}\)
2. b) \(\frac{2}{15}\)
3. d) \(\frac{2}{3}\)
4. d) \(\frac{1}{4}\)