Question

using tree diagrams
a spinner contains 3 equal sections: one red, one yellow, and one blue. the possible results of two spins are shown in the tree diagram below.
what is ( p(\text{red, then yellow}) )?
( \frac{1}{9} )
( \frac{2}{9} )
( \frac{1}{3} )
( \frac{2}{3} )

Explanation:

Step1: Calculate the probability of spinning red first

The spinner has 3 equal - sections. The probability of spinning red (\(P(R)\)) is \(P(R)=\frac{1}{3}\) since there is 1 red section out of 3 total sections.

Step2: Calculate the probability of spinning yellow second

Since the spins are independent events (the result of the first spin does not affect the second spin), the probability of spinning yellow (\(P(Y)\)) is \(P(Y)=\frac{1}{3}\) (1 yellow section out of 3 total sections).

Step3: Use the formula for the probability of independent events

For two independent events \(A\) and \(B\), \(P(A\cap B)=P(A)\times P(B)\). Here, \(A\) is spinning red and \(B\) is spinning yellow. So \(P(\text{red, then yellow})=P(R)\times P(Y)\).
Substitute \(P(R)=\frac{1}{3}\) and \(P(Y)=\frac{1}{3}\) into the formula: \(P(\text{red, then yellow})=\frac{1}{3}\times\frac{1}{3}=\frac{1}{9}\)

Answer:

\(\frac{1}{9}\) (corresponds to the first option)