Idiot\'s Guides Basic Math and Pre-Algebra

256 Part 4: The State of the World

Probabilities with “And”

Calculating the probability of a single card drawn at random from a standard deck being a 7

is fairly simple. There are four 7s out of 52 cards, so the probability is 524 or 131. Calculating

the probability of a particular poker hand is much more complicated, because more cards are

involved, and both the number of successes and the number of cards in the deck change as the

deal goes on. That calculation begins, however, with finding the probability that you get two

cards, two particular cards, one after the other. Let’s say you want a heart and then a diamond.

That’s drawing two cards, of course, not one. So how many ways are there to draw two cards

from a deck of 52? That’s the permutations of 52 things taken 2 at a time, or 52 v 51 = 2,652.

And how many ways to get a heart and then a diamond? Go back to your basic counting

principle. There are 13 ways to get a heart on the first draw, and there are 13 ways to get a

diamond on the second draw. 13 v 13 = 169 ways to get a heart and then a diamond. The

probability of a heart and then a diamond is 169 out of 2,652. Let’s reduce that fraction.

169

2,652

13 13

52 51

13

4 204





 

Another way to think about this is that the probability you get a heart on the first draw is^1352 ,

and the probability of a diamond on the second draw is^1351. (The denominator is only 51 because

you’ve already taken one card out of the deck.)

13

52

13

51

13

4 204

.

Any time you need to find the probability of this event and that one, you want to multiply the

probability of the first event by the probability of the second event. Suppose you roll a die, record

the number that comes up, then roll again and record the second result. What’s the probability

that both of them are even numbers? There are six possibilities for how the die can land, and

three of them are even. The probability of two even numbers is^3

6

3

6

9

36

1

4

.

Sometimes, when you look at the probability of two events occurring in sequence, the results of

the first event have an effect on the probability of the second and sometimes it doesn’t. When

you roll the die twice, the first roll doesn’t affect the second. They’re what we call independent

events. The result of the first roll has no effect on the probabilities for the second.

When you drew the two cards, on the second draw, there would only be 51 cards to choose

from, and that changes the probability. Drawing two cards without replacement is an example

of dependent events, because the result of the first draw changes the probability for the second

draw. But if you choose one card at random from a deck, record what it is, then put it back in

the deck and shuff le before you pull a second card, the probabilities for the second draw are the

same as the first.

If two cards are drawn at random from a standard deck, with replacement—that is, the first card

is drawn, recorded, and replaced before the second card is drawn—the probability of drawing