Question

brenda is rolling a six - sided die and flipping a coin. complete each of the following sentences regarding the probabilities involved in these actions using reduced fractions.
the probability of rolling a five is
the probability of flipping the coin and it landing on heads is
the probability of rolling a five and landing on heads is
are the events independent or dependent?

Explanation:

Step1: Calculate probability of rolling a five

A six - sided die has 6 possible outcomes (\(n = 6\)). Rolling a five is 1 favorable outcome (\(m=1\)). Using the formula \(P=\frac{m}{n}\), the probability of rolling a five is \(\frac{1}{6}\).

Step2: Calculate probability of flipping a coin and landing on heads

A coin has 2 possible outcomes (\(n = 2\)). Landing on heads is 1 favorable outcome (\(m = 1\)). Using the formula \(P=\frac{m}{n}\), the probability of landing on heads is \(\frac{1}{2}\).

Step3: Calculate probability of rolling a five and landing on heads

Since the two events (rolling a die and flipping a coin) are independent, we use the formula \(P(A\cap B)=P(A)\times P(B)\). Here \(P(A)=\frac{1}{6}\) (rolling a five) and \(P(B)=\frac{1}{2}\) (landing on heads). So \(P(A\cap B)=\frac{1}{6}\times\frac{1}{2}=\frac{1}{12}\).

Step4: Determine if events are independent

The outcome of rolling a die does not affect the outcome of flipping a coin. By the definition of independent events (if \(P(A\cap B)=P(A)\times P(B)\) and the occurrence of one event does not influence the occurrence of the other), the events are independent.

Answer:

The probability of rolling a five is \(\frac{1}{6}\). The probability of flipping the coin and it landing on heads is \(\frac{1}{2}\). The probability of rolling a five and landing on heads is \(\frac{1}{12}\). The events are independent.