Question

consider the figures to the right. assume that one number from 1 to 7 is equally likely to be selected at random. each number corresponds to one of the seven figures shown. find the probability of selecting a circle, given that an odd number is selected. p(circle | odd) = (type an integer or a simplified fraction.)

Explanation:

Step1: Determine the number of odd - numbered figures

The odd - numbered figures are \(1\), \(3\), \(5\), \(7\). So, \(n(\text{odd}) = 4\).

Step2: Determine the number of odd - numbered figures that are circles

The odd - numbered figures that are circles are \(1\) and \(7\). So, \(n(\text{circle and odd})=2\).

Step3: Use the formula for conditional probability

The formula for conditional probability is \(P(A|B)=\frac{P(A\cap B)}{P(B)}=\frac{n(A\cap B)}{n(B)}\). Here, \(A\) is the event of selecting a circle and \(B\) is the event of selecting an odd - numbered figure. So, \(P(\text{circle}|\text{odd})=\frac{n(\text{circle and odd})}{n(\text{odd})}\).
Substitute \(n(\text{circle and odd}) = 2\) and \(n(\text{odd}) = 4\) into the formula: \(P(\text{circle}|\text{odd})=\frac{2}{4}=\frac{1}{2}\).

Answer:

\(\frac{1}{2}\)