Probability of 3 Events Calculator

Roll a die, toss a coin, draw a random number, and use this probability of 3 events calculator to determine the:

• Probability of at least one event out of three occurring (union of three events);
• Probability of the intersection of three events (all three happening);
• Probability of exactly one event happening; and
• Probability of none of them occurring.

This probability calculator works for three independent events. Enter the probability of each event as a percentage, or change the unit to decimals. Once you fill in the three fields, the calculator will output the:

• Probability at least one event occurs out of the three:

  P(A ∪ B ∪ C);

• Probability of all three events happening:

  P(A ∩ B ∩ C);

• Probability that exactly one of three events happens:

  P(A ∩ B' ∩ C') + P(A' ∩ B ∩ C') + P(A' ∩ B' ∩ C);

• Probability that none of the events occur:

  P(∅).

Depending on which values you're given, you can also input two numbers into the first section of this calculator and one in the second section.

Remember that each probability has to be a number between 0 and 1 (0% and 100%) inclusive. You can change the unit from percentage to a decimal if you please.

Here are the basic rules of probability:

• Probability takes values between 0 (no chance) and 1 (certain) inclusive.

• Complement Rule (probability that an event doesn't occur): P(A') = 1 - P(A).

• Addition rule: P(A ∪ B) = P(A) + P(B) – P(A ∩ B).

• Multiplication rule: P(A ∩ B) = P(A) × P(B) for independent events.

  P(A ∩ B) = P(A) × P(B | A) = P(B) × P(A | B) for dependent events, where P(B | A) and P(A | B) are the conditional probabilities.

Below you'll find the probability rules used in this probability of 3 events calculator. Use them when you need to calculate the probability of three independent events by hand:

• Multiplication rule - To calculate the probability of the intersection of three independent events, multiply the probabilities of each event together:

  P(A ∩ B ∩ C) = P(A) * P(B) * P(C)

• Addition rule - To find the probability of the union of three events (probability that at least one event occurs of the three), sum the probabilities of three events and subtract their intersections:

  P(A ∪ B ∪ C) = P(A) + P(B) + P(C) - P(A ∩ B) - P(A ∩ C) - P(B ∩ C) + P(A ∩ B ∩ C)

  If we apply the multiplication rule we get the formula for the probability of union of three events only in terms of P(A), P(B), and P(C):

  P(A ∪ B ∪ C) = P(A) + P(B) + P(C) - P(A) * P(B) - P(A) * P(C) - P(B) * P(C) + P(A) * P(B) * P(C)

To calculate the probability of exactly one event happening:

1. We sum the probabilities for three possible scenarios:

   1. A happens, B and C don't;
   2. B happens, A and C don't; and
   3. C happens, A and B don't:

   P(A ∩ B' ∩ C') + P(A' ∩ B ∩ C') + P(A' ∩ B' ∩ C).

2. After applying the multiplication rule we get:

   P(A ∩ B' ∩ C') + P(A' ∩ B ∩ C') + P(A' ∩ B' ∩ C) = P(A) * P(B') * P(C') + P(A') * P(B) * P(C') + P(A') * P(B') * P(C').

3. Now we'll use the complement rule and we get a formula for which we only need P(A), P(B), and P(C):

   P(A) * P(B') * P(C') + P(A') * P(B) * P(C') + P(A') * P(B') * P(C) = P(A) * (1 - P(B)) * (1 - P(C)) + (1 - P(A)) * P(B) * (1 - P(C)) + (1 - P(A)) * (1 - P(B)) * P(C).

To find the chance that none of the three events happen, we use the complement rule for the opposite event, in which at least one event happens (we've already derived the formula for this one):

P(∅) = 1 - (P(A) + P(B) + P(C) - P(A) * P(B) - P(A) * P(C) - P(B) * P(C) + P(A) * P(B) * P(C))

Now you're familiar with so many probability rules! Test your knowledge & intuition by going to Omni's lottery calculator to see if buying a lottery ticket is really worth it!