The spinner is divided into 10 equal sectors labeled A, B, C, D, E, F, G, H, I, and J. The vowels in this set of letters are A, E, and I, so there are 3 vowels out of a total of 10 sectors.
To find the expected number of times a vowel will be spun in 50 spins, we can use the formula for expected value:
\[ \text{Expected number of events} = \text{(Total spins)} \times \text{(Probability of event)} \]
The probability of spinning a vowel is given by the ratio of the number of vowels to the total number of sectors:
\[ \text{Probability of spinning a vowel} = \frac{3}{10} \]
Now, we calculate the expected number of times a vowel will be spun in 50 spins:
\[ \text{Expected number of vowels} = 50 \times \frac{3}{10} = 50 \times 0.3 = 15 \]
Thus, the expected number of times a vowel will be spun is approximately 15 times.