Just kidding– but I am going to Memphis!

Mr. Big Food just walked in and asked, “What is the probability of drawing at least one ace and at least one ❤ in two draws from a fair deck.”

And I asked, “With or without replacement?”

“Without,” he said.

Forget flowers and candy and jewelry, this is the sort of thing that keeps a marriage alive– doing a little probability problem together at the kitchen table. ❤

But back to business. “Without replacement” just means that the card you drew first isn’t put back into the deck, so whatever the probability of the second card is, it’s out of 51, not 52.

The way the problem is set up, it doesn’t matter which of the two cards you draw is a heart and which is an ace. In fact, you could get lucky (!) and draw the ace of hearts on the first draw. What’s the probability of that happening?

1 in 52 or 1/52

But you still have to draw a second card. If you drew the ace of hearts on the first draw, what’s the probability you do not draw the ace of hearts on your second draw? Gee! It’s practically guaranteed! 100%!

1 or 51/51

Drawing the ace of hearts on the first draw is a special case. And it has a parallel. Not drawing the ace of hearts on the first draw, but on the second. Remember, the way the problem is set up, it asks for at least one ace and one heart. (In other words, drawing more than one of either is redundant for the problem.)

So what are the odds of not drawing the ace of hearts on the first draw?

51/52

But drawing it on the second?

1/51

Let’s pause and lay this out in terms of probability.

The probability of drawing the ace of hearts on the first draw (1/52) but not on the second (51/51) =

1/52 X 51/51 = 51/2652

There are 51 pairs of cards out of a possible 2652 pairs of cards that result in drawing an ace of hearts and any other card.

And of drawing the ace of hearts on the second but not the first?

51/52 X 1/51 = 51/2652

Even if you are rusty to probability theory, you can see that the denominator of the fraction is always going to be 52X51 = 2652. That’s how many possible pairs of cards there are. So we’ll wait until the end to reduce fractions.

Keep reading to discover why you should care. 

In addition to the special cases of the ace of hearts which satisfies both conditions, there are two other ways of drawing at least one ace and at least one heart: draw a heart and then an ace and vise-versa. But remember, we have already accounted for the odds of the two special cases occurring– we need to take care to not double count.

What is the probability of drawing a non-heart ace on the first draw, and a non-ace heart on the second?

The probability of drawing an ace that is not the ace of hearts on the first draw is 3/52.

The probability of drawing a heart that is not the ace of hearts on the second draw is 12/51.

3/52 X 12/51 = 36/2652

And the probability of drawing a non-ace heart on the first draw (12/52), followed by any ace except the ace of hearts on the second (3/51) is

12/52 X 3/51 = 36/2652

Those were the probabilities for each of only four possible scenarios which would satisfy the conditions of drawing at least one heart and at least one ace in two draws from a fair deck without replacement. The question, though, doesn’t ask for the probability of each of four scenarios. We want the total probability. To determine that, we simply sum the four probabilities:

51/2652 + 51/2652 + 36/2652 + 36/2652 = 174/2652

(See why we didn’t reduce the fractions? What a mess that would have been!)

When all is said and done the probability is about 0.065– 6.5%.

Under the specified conditions, you’ll draw an ace and a heart roughly (see 0.38… comment below) 13 out of every 200 times you draw two cards from the deck.

You may be asking yourself, “Why on God’s Green Earth do I care about this?” (If it’s any consolation, Missy and Rocky tuned out before the page break. BUT if you are taking Mr. Big Food’s Honors Business Ethics course, you may benefit. This is a homework problem on your next assignment.)

Here’s why.

You just never know when someone on the street is going to walk up to you and say, “Hey, Buddy! I’ll give you 15 to 1 odds you can draw an ace and a heart from this here fair deck in two draws.” TAKE IT! Over the course of 100 plays you will on average win $4.00.

If this fellow offers 14:1 odds just laugh and walk away.

A dead fair bet would be somewhere between $14.38 and $14.39.

And now for the quote of the day.

“It’s that 38 point three eight that keeps screwing us up.”

Indeed.

As I commented in the beginning, nothing binds the bonds of love more than a little problem in probability. I don’t care what Pat says.