Sure! Let's find the probability of picking a button that is not blue in a detailed, step-by-step manner.
1. Determine the Total Number of Buttons:
The bag contains buttons of three different colors:
- Red buttons: 30
- Blue buttons: 40
- White buttons: 50
To find the total number of buttons, we add these quantities together:
[tex]\[
\text{Total buttons} = 30 + 40 + 50 = 120
\][/tex]
2. Calculate the Probability of Picking a Blue Button:
The probability of an event is given by the ratio of the number of favorable outcomes to the total number of possible outcomes. In this case, the number of favorable outcomes for picking a blue button is 40.
Hence, the probability of picking a blue button is:
[tex]\[
P(\text{blue}) = \frac{\text{Number of blue buttons}}{\text{Total number of buttons}} = \frac{40}{120} = \frac{1}{3}
\][/tex]
3. Calculate the Probability of Not Picking a Blue Button:
The probability of the complementary event (not picking a blue button) can be found by subtracting the probability of picking a blue button from 1. This is because the sum of the probabilities of an event and its complement is always 1.
[tex]\[
P(\text{not blue}) = 1 - P(\text{blue})
\][/tex]
Substituting the value we found for [tex]\( P(\text{blue}) \)[/tex]:
[tex]\[
P(\text{not blue}) = 1 - \frac{1}{3} = \frac{3}{3} - \frac{1}{3} = \frac{2}{3}
\][/tex]
Thus, the probability that the button picked at random is not blue is:
[tex]\[
P(\text{not blue}) = \frac{2}{3}
\][/tex]
