An Introduction to Calculating Conditional Probability in Mathematics


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Roll the dice and calculate probability: Image by fyuryu

Calculating Combined Probabilities Might Drop Your Chances to Zero

It is possible that determining the prior probability could show that the conditional probability has dropped to zero.

Look again at the spreadsheet. Suppose that, rather than “5 or 6”, the 1st die roll is ‘2’; let’s call this event ‘C’. Then “A ∩ C” is an empty set: since “9-2 = 7”, one cannot attain the sum of 9-12 with two dice if either’s value is ‘2’. So “P(A|C) = P(A  rather than “5 or 6″ C) / P(C) = zero / (1/6) = zero.”

The Case Where P(B) is Zero

Since “P(A|B) = P(A ∩ B) / P(B)”, there is an obvious problem if “P(B) = zero”. Let’s consider what that would mean, in the real world.

To say that the probability of an event is zero is to say that this event is impossible; it cannot occur. How, then, is it meaningful to determine the probability of an event that is conditional on an impossibility?

How Useful is Conditional Probability?

Blackjack offers a real-world example of calculating conditional probability: Image by Laineys Repertoire

Perhaps the most famous use of conditional probability was developed by “card counters” to win at blackjack. This multi-player card game is also called “21”, because the winning hand has playing cards with a total value closest to 21 without exceeding it. Each numeric playing card has its own value; face cards are worth 10 points, and the ace is worth 1 or 11, depending on the player’s choice at any time. So each of the “10”, jack, queen and king cards is worth 10 points. If played with only one full deck of 52 cards, then the probability that the first card dealt is worth 10 points is 16/52.

The player may choose to “stand” with his current hand, or to receive another card, until the point value exceeds 21. The dealer and each player see the cards as they are dealt.

Let’s consider a simplified strategic question: whether to “stand” when the hand’s value is 15, or to draw another card. If most of the high-value cards have already been dealt, the likelihood increases that the next card will have a low value, so “draw another” might be the better choice.

Effectively, the player considers the chances of receiving a high or low card, given the previous cards that have already been dealt.

It may be possible, as in the Blackjack example above, to calculate a new probability function for remaining events while the situation unfolds. However, it is often more simple to use the conditional probability formula introduced near the start of the article.

Understanding Probability Based on Conditions

Conditional probability allows us to calculate probability, or the chance that an event will occur, when a change in one part of the event alters the outcome. These calculations are not as complex as you might think, so don’t be afraid to experiment!


Weisstein, Eric W. Conditional Probability; Independent Events; Independent Statistics; Event. MathWorld-A Wolfram Web Resource. (1999-2012). Accessed January 13, 2012.

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