A standard deck of cards contains 52 cards, with 13 cards in each suit (hearts, diamonds, clubs, spades). Therefore, the probability of drawing a diamond in a single draw is:
\[ P(\text{diamond}) = \frac{\text{number of diamonds}}{\text{total number of cards}} = \frac{13}{52} = \frac{1}{4} \]
Michael draws a card 20 times, and we want to find the expected number of times he draws a diamond. The expected value \(E\) for a binomial distribution can be calculated using the formula:
\[ E = n \cdot p \]
where \(n\) is the number of trials (draws) and \(p\) is the probability of success (drawing a diamond).
In this case, we have:
- \(n = 20\) (the number of draws)
- \(p = \frac{1}{4}\) (the probability of drawing a diamond)
Now we can calculate the expected number of diamonds drawn:
\[ E = 20 \cdot \frac{1}{4} = 5 \]
Therefore, Michael can expect to draw a diamond 5 times.