Probability and Statistics > Probability > Probability of Two Events occurring Together

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## Probability of Two Events Occurring Together: Overview

Answering probability questions can *seem* tricky, but they all really boil down to two things:

- Figuring out if you multiply or add probabilities.
- Figuring out if your events are dependent (one event has an influence on the other) or independent (they have no effect on each other)

### Should I multiply or add probabilities?

You would **add** probabilities if you want to find out if one event *or* another could happen. For example, if you roll a die, and you wanted to know the probability of rolling a 1 **or **a 6, then you would add the probabilities:

Probability of rolling a 1: 1/6

Probability of rolling a 6: 1/6

So the probability of rolling a 1 or a 6 is 1/6 + 1/6 = 2/6 = 1/3.

The probability of both events happening together on the same die is zero, at least with a single throw. But if you wanted to know the probability of rolling a 1 **and then **rolling a 6, that’s when you would multiply (the probability would be 1/6 * 1/6 = 1/36).

**Learning when to add or multiply can get really confusing! ** The best way to learn when to add and when to multiply is to work out as many probability problems as you can. But, in general:

**If you have “or” in the wording, add the probabilities.
If you have “and” in the wording, multiply the probabilities.**

This is just a general rule — there will be exceptions!

### Dependent vs. Independent

Dependent events are connected to each other. For example:

- In order to win at Monopoly, you have to play the game
- In order to find a parking space, you have to drive
- Choosing two cards from a standard deck without replacement (the first choice has a 1/52 chance, the second a 1/51).

**Independent events** aren’t connected; the probability of one happening has no effect on the other. For example:

- playing Monopoly isn’t connected to winning at Scrabble
- Winning the lottery isn’t connected to you winning at Monopoly
- Choosing a card and then rolling a die are not connected

If you aren’t sure about the difference between independent and dependent events, you may want to read this article first:

Dependent or Independent event? how to Tell.

**Tip:** Look for key phrases in the question that tell you if an event is dependent or not. For example, when you are trying to figure out the probability of two events occurring together and the phrase “Out of this group” or “Of this group…” is included, that tells you the events are **dependent**.

## Probability of Two Events Occurring Together: Independent

Use the specific multiplication rule formula. Just multiply the probability of the first event by the second. For example, if the probability of event A is 2/9 and the probability of event B is 3/9 then the probability of both events happening at the same time is (2/9)*(3/9) = 6/81 = 2/27.

**Sample problem:** The odds of you getting a job you applied for are 45% and the odds of you getting the apartment you applied for are 75%. What is the probability of you getting both the new job *and *the new car?

**Step 1:** *Convert your percentages of the two events to decimals.* In the above example:

45% = **.45.**

75% = **.75.**

**Step 2:** *Multiply the decimals from step 1 together:*

.45 x .65 = **.3375** or 33.75 percent.

The probability of you getting the job and the car is 33.75%

*That’s it!*

## Probability of Two Events Occurring Together: Dependent

The equation you use is slightly different.

P(A and B) = P(A) • P(B|A)

where P(B|A) just means “the probability of B, once A has happened)

**Sample problem:** Eight five % of employees have health insurance. Out of those 85%, 45% had deductibles higher than $1,000. What percentage of people had deductibles higher than $1,000?”

**Step 1:** *Convert your percentages of the two events to decimals.* In the above example:

85% = **.85.**

45% = **.45.**

**Step 2:** *Multiply the decimals from step 1 together:*

.85 x .45 = **.3825** or 38.35 percent.

The probability of someone having a deductible of over $1,000 is 38.35%

*That’s how to find the probability of two events occurring together!*

**Tip:** Sometimes it can help to make a sketch or drawing of the problem to visualize what you are trying to do. The following diagram shows the group of people (85% of the population) and the subgroup (45% of the population), making it more obvious that you should be multiplying (because when you translate 45% of the 85% (have insurance with high deductibles) to math, you get .45 * .85).

If you prefer an online interactive environment to learn R and statistics, this free R Tutorial by Datacamp is a great way to get started. If you're are somewhat comfortable with R and are interested in going deeper into Statistics, try this Statistics with R track.

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this explanation was helpful, although I think that it could go into a bit more depth. It does not explain where, if at all, the $1000 fits into place. Why have it in the question if it does not pertain to the answer.

Angie,

I agree. I added a little more info–hopefully that helps to clarify,

Stephanie

Yeah you are right it could have given more information regarding the problems bur I guess if you know the difference between dependent and independent you should not have a problem.

Also, in the problem you ask “how many people had deductibles higher than $1000,” not “what percentage of people” or “what’s the probability that…”

We can’t answer your questions without knowing how many employees are at the company.

You are correct! Thank you for the correction.

In the section: Probability of Two Events Occurring Together: Independent you used the incorrect numbers from your given problem and messed up the conversion between percentages and decimals

Thanks for spotting that, Nick. It’s now fixed.

I am mujeeb .

I am student of commerce.

Plz send me sure event or certain event formulas with complete solutions.

How do you take into account the duration of the event (rate of occurrence)? It seems that if the events occur in a short time the joint probability would be less than it they occur in a longer period of time