Question

1. the spinner is spun twice. find each probability.

a) $p(\text{square both times})$
b) $p(12, \text{then circle})$
c) $p(\text{square, then even number})$

Explanation:

First, count total sections: 12.
Count squares: 8 sections (2,3,4,5,6,7,8,9).
Count circles: 4 sections (1,10,11,12).
Even numbers: 6 sections (2,4,6,8,10,12).

---

Part a:

Step1: Prob of square once

$P(\text{square}) = \frac{8}{12} = \frac{2}{3}$

Step2: Multiply for two spins

$P(\text{square both times}) = \frac{2}{3} \times \frac{2}{3}$
$= \frac{4}{9}$

---

Part b:

Step1: Prob of rolling 12

$P(12) = \frac{1}{12}$

Step2: Prob of rolling circle

$P(\text{circle}) = \frac{4}{12} = \frac{1}{3}$

Step3: Multiply the two probabilities

$P(12, \text{then circle}) = \frac{1}{12} \times \frac{1}{3}$
$= \frac{1}{36}$

---

Part c:

Step1: Prob of square first

$P(\text{square}) = \frac{8}{12} = \frac{2}{3}$

Step2: Prob of even number

$P(\text{even}) = \frac{6}{12} = \frac{1}{2}$

Step3: Multiply the two probabilities

$P(\text{square, then even}) = \frac{2}{3} \times \frac{1}{2}$
$= \frac{1}{3}$

Answer:

a) $\frac{4}{9}$
b) $\frac{1}{36}$
c) $\frac{1}{3}$