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What is the Law of Probability?

What is the Law of Probability? -magoosh

I like to play gin rummy, so I am always thinking about the probability of getting specific cards. Probability is a fantastic thing for prediction but it can be a little messy to figure those predictions too. Fortunately, there are a few basic principles (or laws) that help figure those probabilities out. Let’s investigate some of the basic laws of probability using a standard 52-card deck.

Law of Large Numbers

The law of large numbers is the principle that the more trials you have in an experiment, the closer you get to an accurate value in probability. For example, let’s say I need a face card to complete a meld in my round of gin rummy.

There are four sets of 3 face cards in a standard deck of cards, this means that the basic probability of drawing a face card (provided the cards are replaced) is

P(Face) = 12 ÷ 52 = 0.23

Now, if you draw just one or two cards, you may not get a face card and think “John was totally wrong!” But if you draw a card 1,000 times, you will closer to the actual percentage you calculated. So, the more trials you have, the closer you get to the theoretical probability.

Addition Rule

Now, suppose I want to finish the game on the next turn, but I need a spade or a face card. In this case, we use the addition rule.

Now the formula addition rule is stated as

P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

This means that we add the probabilities of each event occurring and then subtract the overlap. You don’t want to count those twice. Our set up looks like

P(Face or Spade) = P(Face) + P(Spade) – P(Face ∩ Spade)

And given that there are 12 face cards, 13 spades, and 3 cards that are a face and a spade, our setup up looks like this

P(Face or Spade) = (12 / 52) + (13 / 52) – (3 / 52)

P(Face or Spade) = 0.23 + 0.25 – 0.06

P(Face or Spade) = 0.43

Multiplication Rule

The multiplication rule is one that deals with the case of and in probabilities. It means that the probability of two separate events occurring is the product of each event occurring. The multiplication rule deals most closely with the intersection of two sets. Formally, the rule is stated as

P(A and B) = P(A) ∙ P(B|A) = P(A ∩ B)

So let’s say that I need a spade and a face card in order to go gin in my game and get bonus points. This would mean that I need to figure the product probability.

P(Spade and Face) = P(Spade) ∙ P(Face)

P(Spade and Face) = (12 / 52) ∙ (13 / 52)

P(Spade and Face) = 0.23 ∙ 0.25

P(Spade and Face) = 0.06

The Takeaways

The law of probability tells us about the probability of specific events occurring. The law of large numbers states that the more trials you have in an experiment, then the closer you get to an accurate probability. The addition rule deals with the case of or in the probability of events occurring. The multiplication rule deals with the case of and in the probability of two events occurring together. I hope to see any questions that you have below. Happy statistics!

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