Using Reason to Calculate Probability

Instructor: Michael Quist

Michael has taught college-level mathematics and sociology; high school math, history, science, and speech/drama; and has a doctorate in education.

Probabilities can get pretty complicated, but often you can get to the answer just by thinking it out. In this lesson, we'll explore the use of reason and logic to get answers in basic probability situations.

What are Probabilities?

You're playing a dice game with your friend, John. A pair of doubles wins for you, non-doubles rolls less than 7 win for him. Anything else, roll again. Suddenly, John looks up, angrily. ''Wait a minute! This game isn't fair!'' Is he right?

The probability of a certain event is the chance that it will happen, generally expressed as a number between 0 and 1. The numeric probability can be converted to a percent chance by multiplying by 100%. A 1 probability is a 100% chance, a .5 probability is a 50% chance, and a 0 means a 0% chance.

Some probabilities are almost certainties, very close to 1. For example, the probability that you will take your next breath or that the sun will appear tomorrow morning (unless you happen to be drowning or are in the extreme north or south during the 'long night' season) are both very close to certain. Other events aren't quite so certain. For example, if you flip an equally-weighted coin into the air and allow it to land on the floor, there is a .5 (50%) chance for heads and the same chance for tails.

Calculating Probabilities

So how do we calculate probabilities? The probability of any event happening in dice, cards, coins, or other similar random events is just a matter of dividing the number of possible wins by the number of possible events.

Probability = Number of Winning Combinations / Number of Possible Outcomes

Using logical thinking, or reasoning, we can often determine the probability of success. Let's see if we can figure out whether the dice game is fair. Sometimes the simplest way to figure out a probability is to fill out a table that shows all of the possibilities, and then count the number of ways to win, out of the number of possible outcomes.

Dice possibilities
dice probabilities

Looking at the table, we can see that there are six possible results on the first die, six possible results on the second, and 36 (or 6 x 6) possible combinations between them. Now, the doubles happen when both dice have the same number. On the table, this happens six times.

In the right column you can see the possibilities for John to win. There are quite a few ways to get less than 7 on two dice! Even when you pull out the sets of doubles that are less than 7 (double 1s, double 2s, and double 3s) there are still twice as many as the ways for John to win! The probability for you to win turns out to be 6/36, or just under 17%. John wins in 12 out of 36 possibilities, or about 33% of the time. He was right. The game wasn't fair, but he was the one with the advantage.


Let's try to reason out another one. Say you have a regular deck of playing cards:

  • Four suits of 13 cards each, ♥, ♦, ♠, and ♣
  • In each suit, there is a 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, king, and an ace, where the ace is the highest and the 2 is the lowest

The game is simple. You each draw a random card from the deck, and the highest card wins. If you tie, then you both draw again. John jumps right in and grabs a card. It's a 10.

Being a recent student of probabilities, you calculate the chance that your card will beat his. You know there are 51 cards left in the deck, since he took one. And only jacks, queens, kings, and aces will beat that 10. There are four suits with those four cards, so 16 cards in the deck will beat a 10. Your chances of beating his card are 16/51, or about 31%. Ouch! Not even a 1 in 3 chance!

John laughs when he sees your face and hears your calculation. Then he offers you a choice between two options: if you draw a 10, he'll also call that a win, or you may draw twice and use the higher of the two cards you draw against his. Which choice has a better probability of winning?

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