Elizabeth is going to roll a fair $6-$ sided die $600$ times. $\newline$ Complete the following statement with the best prediction. $\newline$ Elizabeth will roll an even number... $\newline$ Choose $1$ answer: $\newline$ (A) Exactly $100$ times $\newline$ (B) Close to $100$ times but probably not exactly $100$ times $\newline$ (C) Exactly $300$ times $\newline$ (D) Close to $300$ times but probably not exactly $300$ times

Full solution

Q. Elizabeth is going to roll a fair

6-

sided die

600

times.

\newline

Complete the following statement with the best prediction.

\newline

Elizabeth will roll an even number...

\newline

Choose

1

answer:

\newline

(A) Exactly

100

times

\newline

(B) Close to

100

times but probably not exactly

100

times

\newline

300

times

\newline

(D) Close to

300

times but probably not exactly

300

times

Fair Die Probability: Since the die is fair, each of the six faces has an equal probability of $\frac{1}{6}$ of being rolled. There are three even numbers on a die ( $2$ , $4$ , and $6$ ), so the probability of rolling an even number is $\frac{3}{6}$ , which simplifies to $\frac{1}{2}$ .
Calculate Expected Even Rolls: To predict how many times an even number will be rolled in $600$ rolls, we multiply the total number of rolls by the probability of rolling an even number. This gives us $600 \times (\frac{1}{2})$ .
Calculation Result: Performing the calculation, we get $600 \times \left(\frac{1}{2}\right) = 300$ . This means that, on average, Elizabeth should roll an even number $300$ times out of $600$ rolls.
Variability in Even Rolls: However, while the expected value is $300$ , it is unlikely that Elizabeth will roll an even number exactly $300$ times due to the variability inherent in random events. The actual number of even rolls will likely be close to $300$ , but not exactly $300$ .