Question

suppose the spinner shown to the right is spun once, to determine a single - digit number, and we are interested in the event e that the resulting number is odd. assume all sectors of the spinner are the same size.
determine each of the following.
(a) the sample space
(b) the number of favorable outcomes
(c) the number of unfavorable outcomes
(d) the total number of possible outcomes
(e) the probability of an odd number
(f) the odds in favor of an odd number

Explanation:

(a) Sample Space

The sample space is the set of all possible outcomes. Since it's a single - digit number, the sample space \(S=\{0,1,2,3,4,5,6,7,8,9\}\).

(b) Number of Favorable Outcomes

Favorable outcomes (odd numbers). Odd single - digit numbers are \(1,3,5,7,9\). So the number of favorable outcomes \(n(E)=5\).

(c) Number of Unfavorable Outcomes

Unfavorable outcomes (even numbers). Even single - digit numbers are \(0,2,4,6,8\). So the number of unfavorable outcomes \(n(\overline{E}) = 5\).

(d) Total Number of Possible Outcomes

The total number of single - digit numbers is \(N = 10\).

(e) Probability of an Odd Number

The probability formula is \(P(E)=\frac{n(E)}{N}\).
Substitute \(n(E) = 5\) and \(N=10\) into the formula:
\(P(E)=\frac{5}{10}=\frac{1}{2}\).

(f) Odds in Favor of an Odd Number

The odds in favor formula is \(O(E)=\frac{n(E)}{n(\overline{E})}\).
Substitute \(n(E) = 5\) and \(n(\overline{E})=5\) into the formula:
\(O(E)=\frac{5}{5}=1:1\).

Answer:

(a) \(S = \{0,1,2,3,4,5,6,7,8,9\}\)
(b) \(5\)
(c) \(5\)
(d) \(10\)
(e) \(\frac{1}{2}\)
(f) \(1:1\)