Question

What is the average number of tosses required to get a 4 when rolling a fair 10-sided die?

Answer 1

To find the average number of tosses required to get a 4 when rolling a fair 10-sided die, we can model this situation using the concept of expected value.

A fair 10-sided die has outcomes from 1 to 10, and the probability of rolling a 4 is:

\[
P(4) = \frac{1}{10}
\]

The probability of not rolling a 4 in a single toss is:

\[
P(\text{not } 4) = 1 - P(4) = 1 - \frac{1}{10} = \frac{9}{10}
\]

Let \( X \) be the random variable representing the number of tosses until we roll a 4. The situation can be modeled as a geometric distribution where the success (rolling a 4) occurs with probability \( p = \frac{1}{10} \).

The expected value \( E(X) \) for a geometrically distributed random variable is given by:

\[
E(X) = \frac{1}{p}
\]

Substituting \( p \) with \( \frac{1}{10} \):

\[
E(X) = \frac{1}{\frac{1}{10}} = 10
\]

Thus, the average number of tosses required to get a 4 when rolling a fair 10-sided die is:

\[
\boxed{10}
\]