To find the probability that a button picked at random from the bag is blue or not red, we will proceed step-by-step.
1. Find the total number of buttons in the bag:
The bag contains red, blue, and white buttons:
- Number of red buttons: 30
- Number of blue buttons: 40
- Number of white buttons: 50
The total number of buttons is:
[tex]\[
\text{Total buttons} = 30 + 40 + 50 = 120
\][/tex]
2. Calculate the probability of picking a blue button:
The number of blue buttons is 40. The probability of picking one blue button out of the total 120 buttons is:
[tex]\[
P(\text{blue}) = \frac{\text{Number of blue buttons}}{\text{Total number of buttons}} = \frac{40}{120} = \frac{1}{3}
\][/tex]
So, the probability of picking a blue button is [tex]\( \frac{1}{3} \)[/tex] or approximately 0.3333 (33.33%).
3. Calculate the probability of picking a button that is not red:
The non-red buttons are blue and white. The number of blue buttons is 40 and the number of white buttons is 50. Therefore, the total number of non-red buttons is:
[tex]\[
\text{Number of non-red buttons} = 40 + 50 = 90
\][/tex]
The probability of picking a non-red button out of the total 120 buttons is:
[tex]\[
P(\text{not red}) = \frac{\text{Number of non-red buttons}}{\text{Total number of buttons}} = \frac{90}{120} = \frac{3}{4}
\][/tex]
So, the probability of picking a non-red button is [tex]\( \frac{3}{4} \)[/tex] or 0.75 (75%).
4. Determine the probability of picking a blue button or a button that is not red:
Since the set of blue buttons is entirely contained within the set of non-red buttons, picking a blue button is essentially a subset of picking a non-red button. Therefore:
[tex]\[
P(\text{blue or not red}) = P(\text{not red}) = \frac{3}{4}
\][/tex]
To summarize, the probability that the button picked is blue or is not red is:
[tex]\[
P(\text{blue or not red}) = \frac{3}{4}
\][/tex]
So, we conclude that the probability of picking a blue button or a button that is not red is [tex]\(\frac{3}{4}\)[/tex] or 0.75.
