To determine how many times Damian is expected to pull a vanilla-flavored candy from the jar, we first need to calculate the total number of candies in the jar.
The total number of candies is the sum of the frequencies of each flavor:
\[ \text{Total candies} = 51 , (\text{Chocolate}) + 40 , (\text{Fudge}) + 14 , (\text{Orange}) + 20 , (\text{Vanilla}) = 125 \]
Next, we need to find the probability of pulling a vanilla-flavored candy. The probability \(P\) of pulling a vanilla candy is given by the frequency of vanilla candies divided by the total number of candies:
\[ P(\text{Vanilla}) = \frac{\text{Frequency of Vanilla}}{\text{Total candies}} = \frac{20}{125} = \frac{4}{25} \]
Now, if Damian pulls a piece of candy 50 times, the expected number of times he pulls a vanilla flavor candy can be calculated by multiplying the probability of pulling a vanilla candy by the number of pulls:
\[ \text{Expected pulls of Vanilla} = P(\text{Vanilla}) \times \text{Total pulls} = \frac{4}{25} \times 50 = 8 \]
Thus, the expected number of times a vanilla-flavored candy will be pulled from the jar is 8.