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The "classic" high school probability exercises most of the time ends up in a tree of possibilities that deals with all the scenarios (combinations?) that are possible for a given scenario: be it coin tosses, balls in a jar, anagrams, etc.

But there is something that I always have a had time deciding: when to add and when to multiply quantities under a certain scenario? I see that for each branch of the tree, I can just multiply, but when does addition enters the picture?

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Draconar
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4 Answers4

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Roughly speaking (not always 100% true!), in probability, the word or translates into addition, while and translates into multiplication.

The added assumptions are:

  • you can only add if the two events are disjoint
  • you can only multiply if the two events are independent.

For example, if I throw one fair $6$ sided die, the probability of rolling $5$ or more is equal to the probability of roling $5$ or $6$. The probability of that is (given that the events cannot happen at the same time) the probability of rolling a $5$ plus the probability of rolling a $6$, meaning $\frac16+\frac16=\frac13.$

On the other hand, if I throw a die twice, the probability of rolling $6$ both times is the probability of rolling $6$ in the first attempt and rolling $6$ in the second attempt. The events are independent (the second roll is not affected by the first), do the probability of rolling $6$ in both attempts is the probability of rolling $6$ in the first attempt times the pribability of rolling $6$ in the second attempt, meaning $\frac16\cdot\frac16=\frac1{36}.$

5xum
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Add for (disjoint) unions, multiply for (independent) intersections.

Did
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You multiply when you go down a branch (thinning it out); you add when you gather branches together.

The chance of heads this flip followed by tails next flip is $\frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}$; the chance of either heads or tails this flip is $\frac{1}{2} + \frac{1}{2} = 1$.

If you're confused about whether to add or multiply two probabilities, imagine that they're both $\frac{1}{2}$ and think about whether the result should be $\frac{1}{4}$ or $1$. It'll usually be obvious.

dfan
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This is how I try to think of it from non math perspective. Not sure how accurate this is.

If you think that a compounding event will increase your chance compared to the single event, you use the addition rule. Examples:

  • Probability of rolling 4 or bigger in a fair dice. (The OR generally increase your chance as well). The single event is rolling a number(1/6). The compounding events are rolling 4 or bigger. Logically, you will have more chance. You add.
  • Probability of getting Number cards from a deck of card. You calculate this using 1/52 probability for each card. Generally you would get a bigger chance if the ask is to drawANY number. Therefore, you add

However, if you think compounding events will decrease your chance compared to the single event, use the multiplication rule. Most of the time you will see the word AND indicating 2 events happening together

Example:

  • Probability of getting 2 sixes when rolling a dice 2 times( 6 AND 6). The single event of rolling a six is 1/6. If you need to roll another six the second time, logically, this will be harder. Multiply

  • Probability of drawing 3 Aces in a row. This is harder than just drawing 1 ACE. multiply.

user2773013
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