A lady carries lipstick tubes in a bag in her purse. The probability of pulling out the color she wants is $\dfrac{2}{3}$. Suppose she uses her lipstick $4$ times a day. Find the probability
- P(never the correct color)
- P(no more than $3$ times correct)
Let $p$ be the probability of selecting the right color lipstick.
Let $p$ be the probability of not selecting the right color lipstick.
Given: $\;$ $p = \dfrac{2}{3}$ $\;$ $\therefore \;$ $q = 1 - p = 1 - \dfrac{2}{3} = \dfrac{1}{3}$
Total number of times lipstick is selected $= 4$
- P(never selecting the correct color) $= q^4 = \left(\dfrac{1}{3}\right)^4 = \dfrac{1}{81}$
- P(no more than $3$ times correct)
$\begin{aligned} P\left(\text{no more than 3 times correct}\right) & = 1 - P \left(\text{all times correct}\right)\\\\ & = 1 - \left(p\right)^4 \\\\ & = 1 - \left(\dfrac{2}{3}\right)^4 \\\\ & = 1 - \dfrac{16}{81} = \dfrac{65}{81} \end{aligned}$