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### Course: Statistics and probability > Unit 7

Lesson 1: Basic theoretical probability# Probability: the basics

Explore what probability means and why it's useful.

Probability is simply how likely something is to happen.

Whenever we’re unsure about the outcome of an event, we can talk about the probabilities of certain outcomes—how likely they are. The analysis of events governed by probability is called statistics.

**The best example for understanding probability is flipping a coin:**

There are two possible outcomes—heads or tails.

What’s the probability of the coin landing on Heads? We can find out using the equation $P(H)=?$ .You might intuitively know that the likelihood is half/half, or 50%. But how do we work that out? Probability =

In this case:

**Probability of an event = (# of ways it can happen) / (total number of outcomes)**

P(A) = (# of ways A can happen) / (Total number of outcomes)

**Example 1**

There are six different outcomes.

What’s the probability of rolling a one?

What’s the probability of rolling a one or a six?

Using the formula from above:

What’s the probability of rolling an even number (i.e., rolling a two, four or a six)?

**Tips**

- The probability of an event can only be between 0 and 1 and can also be written as a percentage.
- The probability of event
is often written as$A$ .$P(A)$ - If
, then event$P(A)>P(B)$ has a higher chance of occurring than event$A$ .$B$ - If
, then events$P(A)=P(B)$ and$A$ are equally likely to occur.$B$

**Next step:**

Practice basic probability skills on Khan Academy —try our stack of practice questions with useful hints and answers!

## Want to join the conversation?

- If two standard dice are rolled. What is the probability that the total of two dice is less than 6?(26 votes)
- less than 6 would not include 6 so

[ ]

| 1-1 2-1 3-1 4-1 |

| 1-2 2-2 3-2 |

| 1-3 2-3 |

| 1-4 |

[ ]

Length=10

total =6x6

⁂ p()=10/36(13 votes)

- A card is drawn from a standard deck of 52 cards. Find the probability that is

a.) a heart or a face card.

b.) a jack or an ace card

c.) a 10 or a spade.(12 votes)- I am just warning you, I don't know much about cards that much, so my numbers may be off.

a. there are 13 heart cards and 12 face cards (aces aren't faces, right?), of which 3 are repeated, so 13+12-3 = 22/52 = 11/26

b. there are 4 jacks and 4 aces, so 4+4 = 8/52 = 4/26 = 2/13

c. there are 4 tens and 13 spades, and one 10 is repeated, so 4+13-1 = 16/52 = 8/26 = 4/13

I hope that helps!(30 votes)

- Can't you multiply the possibility(fraction) with the the same numerator or denominator to get a different but equivalent answer?

Example: 3/4 chance times 3/3(numerator) equals 9/12. At my school, they say you can multiply fractions with the same numerator/denominator, but I haven't taken probability yet in my grade.(4 votes)- Yes you can multiply probabilities with fractions that are equal to one. We usually want the fraction in the simpliest form though.(5 votes)

- does probability always have to be written like a fraction? How do you know when to write it as a percentage?(2 votes)
- Usually, the question concerning probability should specify if they want either fractions or percentages. Here on KA, you can tell if they're asking for a percentage if you see a % sign by the answer box, while for fractions / decimals a small dialogue box will pop up after you click on the answer box telling you which form to put it in. (I've also seen them state which form to use in italics right after the question.)

Hope this helps!😀(7 votes)

- If there were 3 black dogs,4 brown dogs,and 2 white dog what would happen if You took 2 brown dogs away.(2 votes)
- Um...there would be 7 dogs instead of 9. And there would only be 2 brown dogs now. Which is equal to the number of white dogs. Or is there a more complex reason to this? I don't know. Anyway I hope this helps.(4 votes)

- what is the formula of probability(2 votes)
- Classical Probability (Equally Likely Outcomes):

To find the probability of an event happening, you divide the number of ways the event can happen by the total number of possible outcomes.

Probability of an Event Not Occurring:

If you want to find the probability of an event not happening, you subtract the probability of the event happening from 1.

Probability of Independent Events:

If two events don't affect each other (like flipping a coin twice), you multiply their individual probabilities to find the probability of both events happening.

Probability of Mutually Exclusive Events:

If two events can't happen at the same time (like rolling a die and getting a 1 or a 2), you add their individual probabilities to find the probability of either event happening.

Conditional Probability:

If you want to find the probability of one event happening given that another event has already happened, you divide the probability of both events happening by the probability of the second event happening.(3 votes)

- Ok, I think I get it. So, would the probability of picking a yellow marble be 37.5%? I got 37.5% by turning 3/8 into a percentage. If I'm correct, this is a lot easier than I thought.(2 votes)
- Yes that would be indeed correct!(2 votes)

- 3 red marbles and 3 blue marbles. do not replace first marble in bag before picking again. probability that both marbles are blue(1 vote)
- There are 6 marbles in total, and 3 of them are blue, so the probability that the first marble is blue is 3∕6 = 1∕2

Given that the first marble was blue, there are now 5 marbles left in the bag and 2 of them are blue, and the probability that the second marble is blue as well is 2∕5

So, the probability that both marbles are blue is 1∕2 ∙ 2∕5 = 1∕5(3 votes)

- The mall has a merry-go-round with 12 horses on the outside ring. If 12 people randomly choose those horses, what is the probability they are seated in alphabetical order? I've been stuck on this problem for so long and I have no clue to what is the right way to solve this problem?(2 votes)
- a tire manufacturer advertise, " the median life of our new all-season radial tire is 50,000 miles. An immediate adjustment will be made on any tire that does not last 50,000 miles. You purchased four of these tires. What is the probability that two of the tires will wear out before traveling 50,000 miles?(2 votes)
- Since the median is 50,000, that means that each tire has a 50% chance to reach 50,000 miles (from the definition of median). At this point you have a binomial distribution problem with n = 4, k = 2, and p=q=0.5. Odds of EXACTLY 2 tires failing are therefore 4_C_2*0.5⁴ = 6/16 = 3/8. If you want the odds that 2 or more tires fail, then you would need to add the results for k = 3 and k=4 as well which gives you a probability of 11/16.(5 votes)