Question

use dart board for 10 - 12

1. if a dart hits this target randomly, how many equally likely outcomes are there?

(what is the sample space?)

1. if a dart hits the target randomly, what is the probability it will hit

the bullseye
(one of the center squares)
any shaded square

1. if a dart hits the target randomly 180 times, how many times would you expect (in theory) to hit

the bullseye
a white square

Explanation:

Step1: Count the total number of squares

Count all the squares in the dart - board. There are \(25\) squares in total. So, the number of equally - likely outcomes (sample space) \(n(S)=25\).

Step2: Calculate the probability of hitting the bullseye

The bullseye (center square) is \(1\) square. The probability \(P(\text{bullseye})=\frac{\text{Number of bullseye squares}}{\text{Total number of squares}}=\frac{1}{25}\)
The number of shaded squares: count them. There are \(13\) shaded squares. The probability \(P(\text{shaded})=\frac{13}{25}\)

Step3: Calculate the expected number of hits

For the bullseye: If the dart is thrown \(N = 180\) times, and the probability of hitting the bullseye \(p=\frac{1}{25}\), then the expected number of hits \(x\) (using the formula \(x = N\times p\)) is \(x=180\times\frac{1}{25}=\frac{180}{25}=7.2\)
For the white squares: The number of white squares is \(25 - 13=12\). The probability of hitting a white square \(p=\frac{12}{25}\). Using the formula \(x = N\times p\) with \(N = 180\), we get \(x=180\times\frac{12}{25}=\frac{2160}{25}=86.4\)

Answer:

1. The sample space (number of equally - likely outcomes) is \(25\).
2. Probability of hitting the bullseye is \(\frac{1}{25}\), probability of hitting a shaded square is \(\frac{13}{25}\).
3. Expected number of times to hit the bullseye: \(7.2\), expected number of times to hit a white square: \(86.4\)