Question

the tree diagram shows the outcomes of rolling a die and flipping a coin. what is the probability of rolling an even number and flipping a head?

Explanation:

Step1: Determine the probability of rolling an even number

A die has 6 faces. The even numbers are 2, 4, 6. So the number of favorable outcomes for rolling an even number is 3. The probability of rolling an even number \(P(E)=\frac{3}{6}=\frac{1}{2}\).

Step2: Determine the probability of flipping a head

A coin has 2 faces (head and tail). The number of favorable outcomes for flipping a head is 1. The probability of flipping a head \(P(H)=\frac{1}{2}\).

Step3: Use the multiplication rule for independent events

Since rolling a die and flipping a coin are independent events, the probability of both events occurring is \(P = P(E)\times P(H)\). Substitute \(P(E)=\frac{1}{2}\) and \(P(H)=\frac{1}{2}\) into the formula: \(P=\frac{1}{2}\times\frac{1}{2}=\frac{1\times1}{2\times2}=\frac{1}{4}\). Also, using the total number of outcomes from the tree - diagram (12). The favorable outcomes (even number and head: (2,H), (4,H), (6,H)) are 3. So \(P=\frac{3}{12}\) (simplifies to \(\frac{1}{4}\)).

Answer:

\(\frac{3}{12}\)