Conditional probability

Here you will learn about conditional probability, including the definition of conditional probability and how to calculate probabilities when there are conditional elements.

Students will first learn about conditional probability as part of statistics and probability in 7 th grade.

What is conditional probability?

Conditional probability is the probability of an event occurring based on the occurrence of another event. Conditional probability is a fundamental aspect of probability theory.

Conditional probability questions often involve picking two objects from a set. This is because once you have picked the first object, the probabilities change for the second pick, based on the outcome of the first pick.

For example,

You have a bag containing 3 red marbles and 4 blue marbles.

The probabilities of picking each color marble are,

If you picked out one red marble, there would be 2 red marbles and 4 blue marbles left.

The probabilities would now be,

If instead, you picked out one blue marble, there would be 3 red marbles and 3 blue marbles left.

The probabilities would now be,

The probabilities are calculated based on what has already occurred.

Event A could be the outcome of the first pick (example, picking a red ball).

Event B could be the outcome of the second pick (example, picking a blue ball after picking a red ball).

Conditional probability P(A|B) represents the probability of A given B has already occurred. In the context of picking objects sequentially from a set, it means calculating the probability of picking a second object (Event B ) under the condition that a first object (Event A ) has already been picked.

The outcome of the first pick or previous event (Event A ) influences the probability of Event B, illustrating the dependence between these events in sequential scenarios.

Remember: The probability of an event happening is calculated using:

What is conditional probability?

Common Core State Standards

How does this relate to 7 th grade math?

• Grade 7 – Statistics and Probability (7.SP.C.8)
  Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.

How to calculate conditional probability

In order to calculate conditional probability:

1. Identify the number of desired outcomes under the condition.
2. Identify the total number of outcomes under the condition.
3. Write the probability.

Conditional probability examples

Example 1: using a Venn diagram

The diagram below shows the results when 25 students were asked whether they can speak French or Spanish.

A student is chosen at random. Find the probability that the student can speak French given that they can speak Spanish.

1. Identify the number of desired outcomes under the condition.

The desired outcome here is that the student speaks French. (Event A )

The condition is that they speak Spanish. (Event B )

You need to find the conditional probability of A given B, which is to identify the number of students who speak French and Spanish.

There are 10 students who speak French and Spanish.

This will be the numerator of our fraction.

2Identify the total number of outcomes under the condition.

The condition is that they speak Spanish, so you need to consider all the students who speak Spanish.

There are 10+4=14 students who speak Spanish.

This will be the denominator of our fraction.

3Write the probability.

Example 2: using a two-way table

The table below shows the hair and eye color of a group of people.

A person from the group is chosen at random. Find the probability that the person has blonde hair given that they have blue eyes.

Example 3: probability of one event given another event has occurred

A bag contains 6 blue balls and 10 yellow balls. One ball is picked and it is yellow. A second ball is then picked. Find the probability that the second ball is yellow given that the first ball was yellow.

Example 4: probability of one event given another event

Lauren has a deck of cards. Lauren draws one card from the deck. The card she draws is an ace. Lauren then picks a second card. Find the probability that the second card is not an ace given the first card was an ace.

Example 5: probability of two events using a tree diagram

There are 12 socks in a drawer. 4 are white and 8 are black. William picks one sock and then picks a second sock. Find the probability that both socks are white.

Identify the number of desired outcomes under the condition.

You can draw a tree diagram to represent the possible outcomes:

Since there are 12 socks to begin with, 4 of them white and 8 black, the probability of picking a white sock on the first pick is \cfrac{4}{12} and the probability of picking a black sock is \cfrac{8}{12}.

The probabilities for the second pick need to be calculated given the condition that the first pick has already occurred.

Write the probability.

Now let’s assume the first sock was black. If a black sock has already been picked there are now 4 white socks and 7 black socks. There are now 11 socks left altogether.

P(\text{Event})=\cfrac{\text{number of desired outcomes}}{\text{total number of outcomes}}

P(\text{White sock})=\cfrac{4}{11}

P(\text{Black sock})=\cfrac{7}{11}

You can now fill the rest of the tree diagram:

To find the joint probability that both socks are white, you use P(A\cap{B})=P(A)\times{P(B)} , which is the multiplication rule of probability.

\begin{aligned}P(\text{White and White})&=\cfrac{4}{12}\times\cfrac{3}{11} \\\\ &=\cfrac{12}{132} \end{aligned}

Example 6: probability of two events using a tree diagram

Karam has a box of chocolates containing 14 milk chocolates and 6 white chocolates. Karam eats 2 chocolates. Find the probability that Karam eats two chocolates of the same type.

Identify the number of desired outcomes under the condition.

You can represent this using a tree diagram.

You need to work out the probabilities for the second chocolate.

Write the probability.

If the first chocolate was white then there are 14 milk chocolates and 5 white chocolates left. There are 19 chocolates left.

P(\text{milk})=\cfrac{14}{19}

P(\text{white})=\cfrac{5}{19}

Putting this information on the tree diagram

To find the probability that he eats two of the same chocolates, you need the probability that he eats two milk chocolates or two white chocolates.

P(\text{milk and milk})=\cfrac{14}{20}\times\cfrac{13}{19}=\cfrac{182}{380}

P(\text{white and white})=\cfrac{6}{20}\times\cfrac{5}{19}=\cfrac{30}{380}

The probability that Karam eats two of the same type of chocolate is

\cfrac{182}{380}+\cfrac{30}{380}=\cfrac{212}{380}

Teaching tips for conditional probability

• Emphasize that conditional probability is the probability of an event occurring given that another event has already occurred.

• Ensure students understand the notation P(A|B) which means the probability of event A occurring given that event B has occurred.

• Relate conditional probability to concepts they have already learned, such as basic probability, independent and dependent events.

• Introduce calculating conditional probability using the formula of conditional probability: P(A|B)=\cfrac{P(A\cap{B})}{P(B)}. Here, P(A\cap{B}) is the probability of the intersection of A and B, and P(B) is the probability of event B happening.

• Incorporate technology by using online resources to simulate probability problems and online probability calculators to solve them.

• Provide examples so that students can differentiate between conditional probability (events happening given other events) and unconditional probability (individual events occurring independently).

Easy mistakes to make

• Not changing the probability for the second pick when picking two objects
  For example, if you have a bag containing 3 blue balls and 7 yellow balls, the probability of picking a blue ball on the first pick is \cfrac{3}{10}.
  
  If a blue ball has been picked on the first pick, then the probability of the event of a blue ball on the second pick is \cfrac{2}{9} because there are 2 blue balls left and 9 balls altogether.

• Incorrect fraction work
  Adding, subtracting,multiplying, dividing, or simplifying fractions incorrectly.

• Not simplifying fractions
  Usually when you have fractions in math, you simplify them. In probability questions, it may be better to leave fractions in their unsimplified form. This is because you often need to add fractions to solve probability problems.
  
  Remember that fractions need a common denominator in order to add them. Check with the question if you are asked to give your final answer as a fraction in its simplified form or not.

• Mixing “and” with “or” probabilities
  An “and” probability is calculated by multiplying the probabilities of two events together:
  
  The probability of picking a milk and another milk chocolate in Example 6 was P(\text{milk and milk})=\cfrac{14}{20}\times\cfrac{13}{19}=\cfrac{182}{380}.
  
  The probability of picking a white and another white chocolate was P(\text{white and white})=\cfrac{6}{20}\times\cfrac{5}{19}=\cfrac{30}{380}.
  
  The probability of picking two milk chocolates or two white chocolates was
  \cfrac{182}{380}+\cfrac{30}{380}=\cfrac{212}{380}.

Related probability lessons

• How to calculate probability
• Probability formula
• Probability scale

The next lessons are

• Compound events
• Probability distribution

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