### Video Transcript

The table shows the results of
rolling a die 78 times. Using this information, how many
times is the number two expected to appear if the die is rolled 234 times?

The table tells us that the number
one was rolled 23 times. Number two was rolled 17 times. There were 15 threes, 10 fours, two
fives, and 11 sixes rolled. We are also told that the die was
rolled a total of 78 times. This can be checked by adding 23,
17, 15, 10, two, and 11.

We are asked to find the expected
number of times the number two would be rolled if the die was rolled 234 times. We recall that the expected value
can be calculated by multiplying the probability of an event occurring by the number
of trials or experiments. Before we can apply this formula,
we need to calculate the experimental probability of rolling a two. The experimental probability of
rolling a two with this die is equal to the number of twos in the table over the
total number of rolls in the table, which is 17 over 78, since there were 17 twos
rolled and the die was rolled a total of 78 times.

To work out the expected number of
twos from 234 rolls, we now apply the expected value formula. We need to multiply the expected
probability by the number of rolls. We therefore need to multiply 17
over 78 by 234. 78 and 234 are both divisible by
78, so we can simplify this calculation as shown. Finally, multiplying 17 by three
gives us 51.

We can therefore conclude that
using the information in the table, we would expect the number two to appear 51
times if the die was rolled a total of 234 times.