The Poker Hand or Don't Call Out the Professor This goes back to a problem given to my statistics class, 50 years ago. You are dealt 5 cards from a standard deck of 52. What are the odds of you having the Queen of Spades? The Professor said to work on it and be prepared to answer the following day. I said I could give the correct answer right then, which I  did. We took most of the next class discussing/arguing about the answer. I may have won the argument but lost the war (grade). What is the correct answer?
 A: There are $52$ cards in the deck, of which (after random shuffling) the first $5$ will be dealt to you.  The Queen of Spades is equally likely to be in any of the $52$ possible positions, so the probability it is dealt to you is 
$5/52$.
A: The total number of ordered hands - that is, five-element sequences of cards, where each card is different - is $52\cdot51\cdot50\cdot49\cdot 48$.
Now: of these sequences, how many contain the Queen of Spades?


*

*There are $1\cdot 51\cdot 50\cdot 49\cdot 48$-many sequences whose first card is the Queen of Spades.

*There are $51\cdot 1\cdot 50\cdot 49\cdot 48$-many sequences whose second card is the Queen of Spades. Note that we begin with $51$, not $52$: since we know the second card is the Queen of Spades, this only leaves $51$ possibilities for the first card (it can be anything but the Queen of Spades).

*Continuing this way, we see: there are $5$ ways for a sequence to contain the Queen of Spades (first, second, third, fourth, or fifth card), and each way corresponds to $51\cdot50\cdot 49\cdot 48$-many distinct sequences.

*So the total number of sequences containing the Queen of Spades is $51\cdot50\cdot49\cdot48\cdot 5$.
Now we compare these two numbers: the relevant probability is $${\mbox{sequences with QoS}\over\mbox{all sequences}}={51\cdot50\cdot49\cdot48\cdot 5\over 52\cdot51\cdot50\cdot49\cdot 48}={5\over 52}.$$

EDIT: looking at JMoravitz' comment, I see I made a silly mistake: the number I calculated above is the probability that you get the Queen of Spades. But the odds are slightly different: think "the odds are $2$ to $1$". When we speak of odds, we're comparing two probabilities - "Yes You Do" vs. "No You Don't" - rather than calculating one of the probabilities on its own.
In this case, you have a $5\over 52$ probability of having the QoS, and a $47\over 52$ probability of not; so the odds of you having it are $5$ to $47$.
Note that these two things are just different ways of phrasing the same information.
A: We seem to be showing various solutions for the probability. Here are two and a half more:
(1) Combinations.
$$\frac{{1 \choose 1}{51 \choose 4}}{{52 \choose 5}} = \frac{5}{52} = 0.09615.$$
(2) This is essentially the same as $P(X=1),$ for a hypergeometric random variable $X$
that counts the red balls among five chosen at random without replacement
from an urn with 1 red ball and 51 non-red ones. In R statistical software:
    phyper(1, 1, 51, 5) returns 0.09615385.
(3)  A simulation in R of a million repeats of the experiment. I have
numbered the 52 cards so that the Queen of Spades is # 1. The result
should be accurate to three places.
m = 10^6;  get.qs = numeric(m)
for (i in 1:m) {
  hand = sample(1:52, 5)
  get.qs[i] = sum(hand==1)  }
mean(get.qs)
## 0.096086

An Assignment: Write "The Professor is always right!" 500 times, or execute
code.
rep("The professor is always right!", 500)

