Question

the spinner to the right is spun twice in succession to determine a two - digit number. the first spin gives the first digit and the second spin gives the second digit. assume all sectors of the spinner are the same size. determine each of the following. (a) the sample space (b) the probability of an even number (c) the probability of a number with no repeated digits (d) the probability of a number greater than 30 (e) the probability of a prime number (a) type the sample space below. {}(use a comma to separate answers as needed.)

Explanation:

Step1: Determine the sample space

The spinner has 3, 2, 4. When spun twice, the possible two - digit numbers are formed by taking the first spin result as the tens digit and the second spin result as the units digit.
The sample space \(S=\{22,23,24,32,33,34,42,43,44\}\)

Step2: Calculate the probability of an even number

An even number has an even units digit. The even numbers in the sample space are \(22,24,32,34,42,44\). The number of elements in the sample space \(n(S) = 9\). The number of even numbers \(n(E)=6\). Using the probability formula \(P(E)=\frac{n(E)}{n(S)}\), we have \(P(E)=\frac{6}{9}=\frac{2}{3}\)

Step3: Calculate the probability of a number with no repeated digits

The numbers with no repeated digits are \(23,24,32,34,42,43\). The number of such numbers \(n(N) = 6\). Using the probability formula \(P(N)=\frac{n(N)}{n(S)}\), we have \(P(N)=\frac{6}{9}=\frac{2}{3}\)

Step4: Calculate the probability of a number greater than 30

The numbers greater than 30 are \(32,33,34,42,43,44\). The number of such numbers \(n(G)=6\). Using the probability formula \(P(G)=\frac{n(G)}{n(S)}\), we have \(P(G)=\frac{6}{9}=\frac{2}{3}\)

Step5: Calculate the probability of a prime number

A prime number has only two distinct positive divisors: 1 and itself. The prime numbers in the sample space are \(23,43\). The number of prime numbers \(n(P)=2\). Using the probability formula \(P(P)=\frac{n(P)}{n(S)}\), we have \(P(P)=\frac{2}{9}\)

Answer:

(a) \(\{22,23,24,32,33,34,42,43,44\}\)
(b) \(\frac{2}{3}\)
(c) \(\frac{2}{3}\)
(d) \(\frac{2}{3}\)
(e) \(\frac{2}{9}\)