Question

Marcus plays a game in which he spins a spinner over and over again. The spinner has 4 equally sized sections labeled E, F, G, and H. Spinning an E six times means the game is over.

Marcus uses a uniform probability model to predict the number of times the spinner will be spun before the letter E appears 6 times.

What is Marcus's prediction for the number of total spins before the letter E appears 6 times?

Responses

10 spins
10 spins

18 spins
18 spins

24 spins
24 spins

30 spins

Answer 1

To find the expected number of spins before the letter E appears 6 times, we can use the concept of expected value in probability.

Each time Marcus spins the spinner, there is a \( \frac{1}{4} \) chance of landing on E and a \( \frac{3}{4} \) chance of landing on F, G, or H. Therefore, the expected number of spins to land on E once can be calculated using the formula for expected value in a geometric distribution:

\[
E = \frac{1}{p}
\]

where \( p \) is the probability of success (landing on E). In this case, \( p = \frac{1}{4} \), so:

\[
E = \frac{1}{\frac{1}{4}} = 4
\]

This means we expect it to take 4 spins to get one E. Since Marcus needs to get E six times, we multiply the expected spins per E by 6:

\[
\text{Expected total spins} = 6 \times 4 = 24
\]

Therefore, Marcus's prediction for the total number of spins before the letter E appears 6 times is:

**24 spins**