Home

Maths

Probability

Conditional probability - Higher

Conditional probability - Higher

Select Lesson

Exam Board

Select an option

Explainer Video

Loading...
Tutor: Alice

Summary

Conditional probability

​​In a nutshell

If you have conditions that directly affect an event, the probabilities of outcomes of that event change. This is known as conditional probability.



Conditional probability

​​Definition

Conditional probability is defined as the likelihood of an event or outcome occurring, based on the occurrence of a previous event or outcome. 


For instance, if you don't know whether a coin is biased or not, and in almost all previous flips the coin has landed heads, you would assume (and be correct in doing so) that the coin is more likely to land heads again.



Formula for conditional probability

For two events AA and BB, to find the probability of AA given that event BB has happened, use the formula:

P(AB)=P(AB)P(B)P(A |B) = \dfrac{P(A \cap B)}{P(B)} 


Note: Remember that P(AB)P(A\cap B) means the probability of AA intersect BB or both AA and BB occurring.


Example

A bag contains six red balls and ten blue balls. Mickey draws two balls without replacement. What is the probability of Mickey drawing 22​ red balls given that he drew a red ball first?


Calculate the probabilities needed in the formula.

P((Red, Red)(Red first))=616×515=30240P(\text{(Red, Red)}\cap \text{(Red first)})=\dfrac{6}{16} \times \dfrac{5}{15}= \dfrac{30}{240}


P(Red first)=616P(\text{Red first})=\dfrac{6}{16}


Substitute values into the formula.

P(Red,RedRed first)=30240616P(\text{Red,Red} |\text{Red first}) = \dfrac{\dfrac{30}{240}}{\dfrac{6}{16}}


Simplify.

1838=18÷38=18×83=13\dfrac{\dfrac{1}{8}}{\dfrac{3}{8}}=\dfrac{1}{8}\div\dfrac{3}{8}=\dfrac{1}{8}\times\dfrac{8}{3}=\underline{\dfrac{1}{3}}​​


Note: To calculate P(Red, RedRed first)P(\text{Red, Red}\cap \text{Red first}), use the fact that the only scenario where both events occur at the same time, happens when two reds are picked meaning that  P(Red, RedRed first)=P(Red, Red)P(\text{Red, Red}\cap \text{Red first})=P(\text{Red, Red}).



Tree diagrams

You can visually display conditional probabilities on tree diagrams:


Maths; Probability; KS4 Year 10; Conditional probability - Higher

​​P(B)P(B)​​

Probability of B occurring.

P(B)P(\overline{B})   

Probability of B not occurring.

P(AB)P(A|B)​​

Probability of A occurring, given B does.

P(AB)P(A\cap B)    

Probability of A and B both occurring.


Using the rules for finding probabilities from tree diagrams, you can obtain the formula above for conditional probability as following the top branch of the tree gives:

P(B)×P(AB)=P(AB)P(B) \times P(A|B) = P(A \cap B)​​


Which rearranges to give:

P(AB)=P(AB)P(B)P(A |B) = \dfrac{P(A \cap B)}{P(B)}​​



Independence

Definition

Two events are independent if the outcome of one event does not affect the outcome of the other.


For two independent events AA​ and BB:

 P(AB)=P(A)P(B) P(A \cap B) = P(A)P(B)


This means that:

P(AB)=P(AB)P(B)=P(A)P(B)P(B)=P(A)P(A|B) = \dfrac{P(A \cap B)}{P(B)} = \dfrac{P(A)P(B)}{P(B)} = P(A)​​


This formula can be used to check for independence. If, and only if, this formula is satisfied, two events AA and BB independent.


Create an account to read the summary

Exercises

Create an account to complete the exercises

FAQs - Frequently Asked Questions

How can I check if two events are independent?

What are independent events?

What is conditional probability?

Beta

I'm Vulpy, your AI study buddy! Let's study together.