A spinner lands on red with probability 1/3 on each independent spin. It is spun 4 times. What is the probability of at least 3 reds?

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Biased spinner successes

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A spinner lands on red with probability $\frac{1}{3}$ on each independent spin. It is spun $4$ times. What is the probability of at least $3$ reds?

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Hint

Use the binomial formula for exactly $3$ reds and for exactly $4$ reds, then add.

Solution

Exactly $3$ reds has probability $4 \times (\frac{1}{3})^{3} \times (\frac{2}{3}) = \frac{8}{81}$ .
Exactly $4$ reds has probability $(\frac{1}{3})^{4} = \frac{1}{81}$ .
Adding gives $\frac{8}{81} + \frac{1}{81} = \frac{9}{81}$ .
The probability is $\frac{1}{9}$ .

Answer: $\frac{1}{9}$