A bag contains 30 red, 40 blue, and 50 white buttons. You pick one button at random. Find the probability that it is blue or not red. \[ P \text { (blue or not red) }=\text { [?] } \]

To find the probability of picking a blue button or a button that is not red, we first calculate the total number of buttons, which is 30 (red) + 40 (blue) + 50 (white) = 120 buttons in total. Next, we know that there are 40 blue buttons and 90 buttons that are not red (40 blue + 50 white). The probability of picking a blue button or any button that is not red can be found using the formula for the probability of the union of two events: \[ P(\text{blue or not red}) = P(\text{blue}) + P(\text{not red}) - P(\text{blue and not red}). \] - \(P(\text{blue}) = \frac{40}{120} = \frac{1}{3}\), - \(P(\text{not red}) = \frac{90}{120} = \frac{3}{4}\), - \(P(\text{blue and not red}) = P(\text{blue}) = \frac{1}{3}\) (since all blue buttons are not red). Now we put it together: \[ P(\text{blue or not red}) = \frac{1}{3} + \frac{3}{4} - \frac{1}{3}. \] To combine these fractions, let's convert them into a common denominator (which would be 12): - \( \frac{1}{3} = \frac{4}{12}\), - \( \frac{3}{4} = \frac{9}{12}\), - \( \frac{1}{3} = \frac{4}{12}\). So substituting these values in: \[ P(\text{blue or not red}) = \frac{4}{12} + \frac{9}{12} - \frac{4}{12} = \frac{9}{12} = \frac{3}{4}. \] Thus, the probability that the button is blue or not red is \[ \boxed{\frac{3}{4}}. \]