Question

sam and erica are playing a board game. they spin a pointer to determine whether to move forward or back. they toss a number cube to determine how many spaces to move. what is the probability of moving forward an even number of spaces?

Explanation:

Step1: Probability of moving forward

The spinner is split into two equal parts (move forward and move back), so the probability of moving forward, \( P(\text{forward}) \), is \( \frac{1}{2} \).

Step2: Probability of even spaces on a number cube

A standard number cube (die) has numbers 1 - 6. The even numbers are 2, 4, 6. So there are 3 even numbers out of 6 total. Thus, the probability of rolling an even number, \( P(\text{even}) \), is \( \frac{3}{6}=\frac{1}{2} \).

Step3: Multiply the two probabilities

Since the two events (moving forward and rolling an even number) are independent, we multiply their probabilities: \( P = P(\text{forward}) \times P(\text{even})=\frac{1}{2}\times\frac{1}{2}=\frac{1}{4} \). Wait, no, wait. Wait, the number of even numbers on a die: 2,4,6 – that's 3 out of 6, so \( \frac{3}{6}=\frac{1}{2} \). Then moving forward is \( \frac{1}{2} \). So \( \frac{1}{2}\times\frac{1}{2}=\frac{1}{4} \)? Wait, no, wait, maybe I made a mistake. Wait, the die: numbers 1,2,3,4,5,6. Even numbers: 2,4,6 (3 numbers). So probability of even is \( \frac{3}{6}=\frac{1}{2} \). Probability of forward is \( \frac{1}{2} \) (since spinner is split into two equal parts: forward and back). So the combined probability is \( \frac{1}{2}\times\frac{1}{2}=\frac{1}{4} \)? Wait, but let's check again. So the two events are independent, so we multiply. So \( \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \)? Wait, no, wait, maybe the die has numbers 1 - 6, so even numbers are 2,4,6 (3), so probability of even is \( \frac{3}{6} = \frac{1}{2} \). Probability of forward is \( \frac{1}{2} \). So multiplying them: \( \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \). Wait, but let's see the options. The options are \( \frac{1}{4} \), \( \frac{1}{2} \), \( \frac{1}{3} \), \( \frac{1}{6} \). So the correct calculation is:

Probability of moving forward: the spinner is divided into two equal sections (green for forward, red for back), so \( P(\text{forward}) = \frac{1}{2} \).

Probability of rolling an even number on a number cube (die): the cube has numbers 1 - 6. Even numbers are 2, 4, 6. So there are 3 even numbers out of 6 total. Thus, \( P(\text{even}) = \frac{3}{6} = \frac{1}{2} \).

Since the two events (spinning forward and rolling an even number) are independent, we multiply their probabilities:

\( P(\text{forward and even}) = P(\text{forward}) \times P(\text{even}) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \). Wait, but wait, maybe the die in the image shows 1, 2, 3, 4? Wait, the image shows a die with 1 and 4 visible, but it's a standard die, so numbers 1 - 6. So assuming it's a standard die, the calculation is as above. So the probability is \( \frac{1}{4} \)? Wait, no, wait, \( \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \). So the answer should be \( \frac{1}{4} \).

Answer:

\(\frac{1}{4}\) (Assuming the options include \(\frac{1}{4}\); from the given options, one of them is \(\frac{1}{4}\), so the correct answer is \(\frac{1}{4}\))