Question

Lynne is rolling a six-sided number cube repeatedly. Her goal is to roll an even number 23 times.

How many times should Lynne expect to have to roll the number cube to result in an even number 23 times?

Enter your answer in the box to correctly complete the statement.

Lynne should expect to have to roll the number cube approximately
times to result in an even number 23 times.

Answer 1

To determine how many rolls Lynne should expect to make to roll an even number 23 times, we first consider the probabilities involved in rolling a six-sided die.

The even numbers on a standard six-sided die (1, 2, 3, 4, 5, 6) are 2, 4, and 6. Therefore, there are 3 even numbers.

The probability \( P(\text{even}) \) of rolling an even number is:

\[
P(\text{even}) = \frac{\text{Number of even outcomes}}{\text{Total outcomes}} = \frac{3}{6} = \frac{1}{2}
\]

The probability of rolling an odd number is also:

\[
P(\text{odd}) = 1 - P(\text{even}) = 1 - \frac{1}{2} = \frac{1}{2}
\]

Let \( X \) be the number of rolls required to get 23 even outcomes. Given that the die rolls are independent, and each roll can be modeled as a Bernoulli trial where the success (rolling an even number) has a probability of \( \frac{1}{2} \), we can use the concept of expected value for the number of trials.

The expected number of trials \( E(N) \) to achieve \( k \) successes (even rolls) in a series of Bernoulli trials is given by:

\[
E(N) = \frac{k}{p}
\]

where \( k \) is the number of successes needed (which is 23 in this case) and \( p \) is the probability of success on each trial (which is \( \frac{1}{2} \)).

Putting the values into the formula:

\[
E(N) = \frac{23}{\frac{1}{2}} = 23 \times 2 = 46
\]

Thus, Lynne should expect to have to roll the number cube approximately \( 46 \) times to result in an even number \( 23 \) times.

So the final answer is:

Lynne should expect to have to roll the number cube approximately **46** times to result in an even number 23 times.