Calculating the expected value of the number of spades in a hand of $18$ drawn from a deck of $52$ poker cards.

Question

I am having troubles with the question:

You have a standard deck of $52$ playing cards ($13$ of each suit). You draw a hand of the top $18$ of them. Spades are one of the four suits. What is the expected value of the number of spades you draw?

For my approach I calculate the individual probabilities for every event of drawing a spades as so:

Let $P_i$ be the probability that $i$ spades drawn.
So naturally calculating the Expected value would as follows: $$\sum_{i =0}^{n = 13}i\cdot P_i$$

However this task is tedious and leaves the question of where the remaining $5$ cards in the hand adds up to the equation.

Am I even thinking in the right direction? Is there a better way to calculate this?

Answer 1

Fortunately, there is. Let $I_j$ be the the indicator of the event that the $j$th draw is a spade. Then $$N = I_1+\dotsb+I_{18}$$ is the total number of spades in our 18 draws. We notice that $$E[I_j] =P(I_j) = \frac{13}{52} = \frac{1}{4}.$$ Hence, by the linearity of expectation we have that $$E[N] = E[I_1+\dotsb+I_{18}] = E[I_1]+\dotsb+E[I_{18}] = 18\cdot\frac{1}{4} = 4.5$$

It turns out that $N$ follows a hypergeometric distribution and so the expectation is well-known: $$E[N] = \frac{18\cdot 13}{52} = 4.5.$$

Answer 2

It is easy to understand.. Since there are 4 suits, Probability of any random card being a spade is $${1\over 4}$$ So the expected number of spades are $$18\times {1\over 4}=4.5$$