Question

a coin has two sides: heads and tails. a die has six sides, numbered 1 through 6. if you flip the coin and roll the die, what is the probability that you flip \heads\ and roll a number less than 4?

\\(\circ\\) \\(\frac{1}{2}\\)

\\(\circ\\) \\(\frac{3}{4}\\)

\\(\circ\\) \\(\frac{1}{4}\\)

\\(\circ\\) \\(\frac{1}{12}\\)

Explanation:

Identify the individual events

The problem involves two independent actions: flipping a coin and rolling a six-sided die. We need to find the probability of both events occurring together.

Calculate the probability of flipping heads

A standard coin has two outcomes: heads (\(H\)) or tails (\(T\)).
The probability of flipping "heads" is:

$$P(\text{heads}) = \frac{1}{2}$$

Calculate the probability of rolling a number less than 4

A standard die has six faces: \(1, 2, 3, 4, 5, 6\).
The outcomes that are strictly less than \(4\) are \(1, 2,\) and \(3\).
There are \(3\) favorable outcomes out of \(6\) total outcomes:

$$P(\text{less than 4}) = \frac{3}{6} = \frac{1}{2}$$

Calculate the compound probability

Since the coin flip and the die roll are independent events, we multiply their individual probabilities to find the joint probability:

$$P(\text{heads and less than 4}) = P(\text{heads}) \times P(\text{less than 4})$$

$$P(\text{heads and less than 4}) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$$

Answer:

• (A) \(\frac{1}{2}\)
• (B) \(\frac{3}{4}\)
• (C) \(\frac{1}{4}\) (Correct answer)
• (D) \(\frac{1}{12}\)