An Introduction to Calculating Conditional Probability in Mathematics


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How to Calculate Conditional Probability

Conditional Sum of Two Dice: Image by Mike DeHaan

Mathematicians show that the conditional probability of event A, assuming that event B has occurred, is  P(A|B).

The calculation is “P(A|B) = P(A ∩ B) / P(B)”. In English, the conditional probability of event ‘A’, assuming event ‘B’ has occurred, is the probability of the intersection of events ‘A’ and ‘B’, divided by the probability of event ‘B’.

Note that P(B), like the probability of any event, cannot be greater than ‘1’. Therefore dividing by P(B) does not decrease the conditional probability. Instead, it (almost) always increases it.

We should develop an example to illustrate the effects of dividing by P(B).

Let’s roll two dice, one after the other; each will display a number from 1 through 6 with equal probability.

Refer to the spreadsheet image above. The sum of the two dice values is a number from 2 through 12. Each of ‘2’ or ’12’ only has a 1/36 probability; ‘7’ occurs 6 times out of 36 possible rolls.

Let event ‘A’ be “The sum of the two dice has a value from 9 through 12 inclusive”. The spreadsheet shows the probability is 11.11%, or one in nine.

Let event ‘B’ be “The 1st die has the value 5 or 6”. By inspecting the table, we count 12 outcomes where the 1st die is 5 or 6.

The table also shows that 7 of the 12 cases where the 1st die is 5 or 6 ends with the sum from 9 through 12. One expects the conditional probability P(A|B) to be 7/12 or 58.33%. Adding the italicized 16.67% + 16.67% + 16.67% + 8.33% = 58.34%; the discrepancy is merely a rounding error.

By the formula, “P(A|B) = P(A ∩ B) / P(B)”. We counted seven outcomes that satisfy “A ∩ B” of the thirty-six possible outcomes; therefore “P(A ∩ B) = 7/36”. Obviously, “P(B) = 2/6 = 1/3” for the 1st die to show 5 or 6.

Thus “P(A|B) = P(A ∩ B) / P(B) = (7/36) / (1/3) = (7/36) * 3 = 7/12 = 58.34%” as had already been found by inspection.

In simple English: If you roll two dice, the chance that the sum will be 9, 10, 11 or 12 is 1/9, or 11.11%. But if your first roll is a 5 or 6, then the odds of achieving a sum of 9-12 improves to 7/36 or 58.34%.

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