Question

a dartboard has 8 equally sized slices numbered from 1 to 8.
some are grey and some are white.
the slices numbered 1 and 5 are grey.
the slices numbered 2, 3, 4, 6, 7, and 8 are white.
a dart is tossed and lands on a slice at random.
let x be the event that the dart lands on a grey slice, and let ( p(x) ) be the
probability of x.
let not x be the event that the dart lands on a slice that is not grey, and let
( p(\text{not }x) ) be the probability of not x.
(a) for each event in the table, check the outcome(s) that are contained in the event. then, in the last column, enter the probability of the event.
(b) subtract.
( 1 - p(x)=square )

Explanation:

Step1: Calculate \(P(X)\)

The formula for probability is \(P = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\).
For event \(X\) (grey slices), the favorable outcomes are \(1\) and \(5\). So the number of favorable outcomes \(n(X)=2\), and the total number of outcomes \(n = 8\).
\(P(X)=\frac{2}{8}=\frac{1}{4}\)

Step2: Calculate \(P(\text{not }X)\)

For event \(\text{not }X\) (white slices), the favorable outcomes are \(2\), \(3\), \(4\), \(6\), \(7\), \(8\). So the number of favorable outcomes \(n(\text{not }X)=6\), and the total number of outcomes \(n = 8\).
\(P(\text{not }X)=\frac{6}{8}=\frac{3}{4}\)

Step3: Calculate \(1 - P(X)\)

Substitute \(P(X)=\frac{1}{4}\) into \(1 - P(X)\)
\(1-\frac{1}{4}=\frac{4 - 1}{4}=\frac{3}{4}\)

Answer:

• \(P(X)=\frac{1}{4}\)
• \(P(\text{not }X)=\frac{3}{4}\)
• \(1 - P(X)=\frac{3}{4}\)