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Lesson Explainer: Operations on Events: Difference Mathematics

In this explainer, we will learn how to find the probability of the difference of two events.

First, recall the operations on events that you have met so far.

Definition: Complement, Intersection, and Union of Events

The operations on events 𝐴 and 𝐡 are as follows, with the shaded area in the Venn diagram representing each operation respectively.

  • The complement of an event 𝐴 is denoted by 𝐴 and contains elements that are not in 𝐴.
  • The intersection of events 𝐴 and 𝐡 is denoted by 𝐴∩𝐡 and contains elements that are in both 𝐴 and 𝐡.
  • The union of events 𝐴 and 𝐡 is denoted by 𝐴βˆͺ𝐡 and contains elements that are in either 𝐴 or 𝐡 or both.

The new operation that we will meet in this explainer is the difference between two events 𝐴 and 𝐡, as detailed in the definition below.

Definition: Difference of Events

The difference between two events 𝐴 and 𝐡 is denoted by π΄βˆ’π΅ and is illustrated by the shaded area in the Venn diagram below. This contains elements that are in 𝐴 but not in 𝐡.

Using our understanding of Venn diagrams, we can derive the formula for the difference between two events.

By considering the area of the shaded region for π΄βˆ’π΅, this is equivalent to the area of 𝐴 minus the area of 𝐴∩𝐡, as seen below.

Therefore, π΄βˆ’π΅=π΄βˆ’(𝐴∩𝐡). We can then use this to derive a formula for the probability of the difference between two events.

Rule for the Probability of the Difference between Two Events

The probability of the difference between two events 𝐴 and 𝐡 is 𝑃(π΄βˆ’π΅)=𝑃(𝐴)βˆ’π‘ƒ(𝐴∩𝐡).

In the following example, we will apply the rule of probability in the definition above in order to find the probability of the difference between two events.

Example 1: Determining the Probability of the Difference between Two Events

Suppose 𝐴 and 𝐡 are two events. Given that 𝑃(𝐴)=0.3 and 𝑃(𝐴∩𝐡)=0.03, determine 𝑃(π΄βˆ’π΅).

Answer

Recall that the formula for the probability of the difference between two events 𝐴 and 𝐡 is 𝑃(π΄βˆ’π΅)=𝑃(𝐴)βˆ’π‘ƒ(𝐴∩𝐡).

By substituting 𝑃(𝐴)=0.3 and 𝑃(𝐴∩𝐡)=0.03 into the formula above, we get 𝑃(π΄βˆ’π΅)=0.3βˆ’0.03=0.27.

Therefore, 𝑃(π΄βˆ’π΅)=0.27.

In the next example, we will apply the rule of probability for the difference between two events in order to determine the probability of an event, where the difference and intersection are known.

Example 2: Determining the Probability of an Event given the Difference and Intersection of Two Events

Suppose that 𝐴 and 𝐡 are two events. Given that 𝑃(π΄βˆ’π΅)=27 and 𝑃(𝐴∩𝐡)=16, determine 𝑃(𝐴).

Answer

The formula for the probability of the difference between two events 𝐴 and 𝐡 is 𝑃(π΄βˆ’π΅)=𝑃(𝐴)βˆ’π‘ƒ(𝐴∩𝐡).

By substituting 𝑃(π΄βˆ’π΅)=27 and 𝑃(𝐴∩𝐡)=16, we get the following equation: 27=𝑃(𝐴)βˆ’16.

If we rearrange to make 𝑃(𝐴) the subject, we get 27+16=𝑃(𝐴)𝑃(𝐴)=12+742=1942.

Therefore, 𝑃(𝐴)=1942.

In the next example, we will consider how to find the probability of the difference between two events given in a context.

Example 3: Finding the Probability of the Difference between Two Events Given in a Context

A ball is drawn at random from a bag containing 12 balls each with a unique number from 1 to 12. Suppose 𝐴 is the event of drawing an odd number and 𝐡 is the event of drawing a prime number. Find 𝑃(π΄βˆ’π΅).

Answer

To find 𝑃(π΄βˆ’π΅), we use the formula for the probability of the difference between two events 𝐴 and 𝐡, which is 𝑃(π΄βˆ’π΅)=𝑃(𝐴)βˆ’π‘ƒ(𝐴∩𝐡).

In order to do this, we must find 𝑃(𝐴) and 𝑃(𝐴∩𝐡).

To find 𝑃(𝐴), we first identify the set 𝐴. We know that 𝐴 is the event of drawing an odd number from a bag with balls numbered from 1 to 12. Therefore, 𝐴 is given by the set {1,3,5,7,9,11}.

Since the number of outcomes in 𝐴 is 6 and the total number of outcomes is 12 (since there are 12 balls in the bag), therefore the probability of 𝐴 is given by 𝑃(𝐴)=𝐴=612=12.numberofoutcomesintotalnumberofoutcomes

In order to find 𝑃(𝐴∩𝐡), we start by identifying the sets 𝐴 and 𝐡, and the set 𝐴∩𝐡. We know that 𝐴 is given by the set {1,3,5,7,9,11} (as stated above). Set 𝐡 is the event of drawing a prime number from a bag with balls numbered from 1 to 12. Therefore, 𝐡 is given by the set {2,3,5,7,11}.

We can see that 𝐴∩𝐡, the intersection of 𝐴 and 𝐡, is the set that contains the elements that occur in both 𝐴 and 𝐡. In this case, 𝐴∩𝐡={3,5,7,11}. The probability of 𝐴∩𝐡 is given by 𝑃(𝐴∩𝐡)=𝐴𝐡=412=13.numberofoutcomesinandtotalnumberofoutcomes

We can now substitute 𝑃(𝐴)=12 and 𝑃(𝐴∩𝐡)=13 into the formula 𝑃(π΄βˆ’π΅)=𝑃(𝐴)βˆ’π‘ƒ(𝐴∩𝐡) in order to find 𝑃(π΄βˆ’π΅): 𝑃(π΄βˆ’π΅)=12βˆ’13=3βˆ’26=16.

Therefore, 𝑃(π΄βˆ’π΅)=16.

Next, we will consider the probability of the difference between two events, where one event is a subset of another.

Recall that if 𝐡 is a subset of 𝐴, then all of the elements in 𝐡 are in 𝐴 and that the intersection of 𝐴 and 𝐡 is 𝐡, or 𝑃(𝐴∩𝐡)=𝑃(𝐡). This can be seen in the Venn diagram below:

We can then use this to find the rules of probability for the difference between two events.

Definition: Probability of the Difference between Two Events, Where One Is a Subset of Another

For two events 𝐴 and 𝐡, where 𝐡 is a subset of 𝐴, denoted π΅βŠ‚π΄, it follows that the probability of the difference between two events is as follows:

  • 𝑃(π΄βˆ’π΅)=𝑃(𝐴)βˆ’π‘ƒ(𝐡), since 𝑃(𝐴∩𝐡)=𝑃(𝐡),
  • 𝑃(π΅βˆ’π΄)=𝑃(𝐡)βˆ’π‘ƒ(𝐡)=0, since 𝑃(𝐴∩𝐡)=𝑃(𝐡).

In the next example, we will use the rules for the probability of the difference between two events when one is a subset of another event.

Example 4: Determining the Probability of an Event given the Difference between Two Events, Where One Is a Subset

Suppose 𝐴 and 𝐡 are two events. Given that π΅βŠ‚π΄, 𝑃(𝐡)=49, and 𝑃(π΄βˆ’π΅)=15, determine 𝑃(𝐴).

Answer

For two events 𝐴 and 𝐡, where π΅βŠ‚π΄, we know that 𝑃(𝐴∩𝐡)=𝑃(𝐡). For the difference of two events, this gives us 𝑃(π΄βˆ’π΅)=𝑃(𝐴)βˆ’π‘ƒ(𝐡).

Since 𝑃(π΄βˆ’π΅)=15 and (𝐴∩𝐡)=𝑃(𝐡)=49, by substitution we can form the equation 15=𝑃(𝐴)βˆ’49.

Therefore, rearranging to make 𝑃(𝐴) the subject gives us 𝑃(𝐴)=15+49=9+2045=2945.

Therefore, 𝑃(𝐴)=2945.

We can use multiple rules of probability for operations on events in order to solve problems. We will next consider two of these rules of probability, the complement and the union of events. Let’s recall what these rules are.

Definition: Rules of Probability for Complement and Union of Events

  • The probability of the complement of an event 𝐴 is 𝑃(𝐴)=1βˆ’π‘ƒ(𝐴).
  • The probability of the union of events 𝐴 and 𝐡 is 𝑃(𝐴βˆͺ𝐡)=𝑃(𝐴)+𝑃(𝐡)βˆ’π‘ƒ(𝐴∩𝐡).

The following example uses the rules of probability for the union of two events and the difference between two events.

Example 5: Determining the Probability of the Difference of Two Events Using the Addition Rule

Suppose 𝐴 and 𝐡 are two events with probabilities 𝑃(𝐴)=57 and 𝑃(𝐡)=47. Given that 𝑃(𝐴βˆͺ𝐡)=67, determine 𝑃(π΄βˆ’π΅).

Answer

As we are required to find 𝑃(π΄βˆ’π΅), we must use the rule of probability for the difference between two events, which is stated as follows: 𝑃(π΄βˆ’π΅)=𝑃(𝐴)βˆ’π‘ƒ(𝐴∩𝐡).

Since we do not know 𝑃(𝐴∩𝐡), we must use a further rule to determine this. As we know 𝑃(𝐴βˆͺ𝐡) as well as 𝑃(𝐴) and 𝑃(𝐡), we can use the addition rule for probability: 𝑃(𝐴βˆͺ𝐡)=𝑃(𝐴)+𝑃(𝐡)βˆ’π‘ƒ(𝐴∩𝐡).

When we substitute 𝑃(𝐴)=57, 𝑃(𝐡)=47, and 𝑃(𝐴βˆͺ𝐡)=67, we get an equation with 𝑃(𝐴∩𝐡): 67=57+47βˆ’π‘ƒ(𝐴∩𝐡).

Rearranging to solve for 𝑃(𝐴∩𝐡) gives us 𝑃(𝐴∩𝐡)+67=57+47𝑃(𝐴∩𝐡)=57+47βˆ’67=37.

Since we have found 𝑃(𝐴∩𝐡)=37, we can substitute this, along with 𝑃(𝐴)=57, into the formula 𝑃(π΄βˆ’π΅)=𝑃(𝐴)βˆ’π‘ƒ(𝐴∩𝐡).

So, by substituting, we can solve for 𝑃(π΄βˆ’π΅): 𝑃(π΄βˆ’π΅)=57βˆ’37=27.

Therefore, 𝑃(π΄βˆ’π΅)=27.

The next example uses the rules of probability for the complement of an event, the union of an event, and the difference between two events.

Example 6: Determining the Probability of the Difference of Two Events Using the Addition Rule and the Complement Rule

Suppose that 𝐴 and 𝐡 are events in a random experiment. Given that 𝑃(𝐴)=0.71, 𝑃(𝐡)=0.47, and 𝑃(𝐴βˆͺ𝐡)=0.99, determine 𝑃(π΅βˆ’π΄).

Answer

As we are required to find 𝑃(π΅βˆ’π΄), we must use the rule of probability for the difference between two events, which is stated as follows: 𝑃(π΄βˆ’π΅)=𝑃(𝐴)βˆ’π‘ƒ(𝐴∩𝐡).

As events 𝐴 and 𝐡 are reversed in this formula, we need to rewrite this as 𝑃(π΅βˆ’π΄)=𝑃(𝐡)βˆ’π‘ƒ(𝐡∩𝐴)=𝑃(𝐡)βˆ’π‘ƒ(𝐴∩𝐡) since 𝑃(𝐡∩𝐴)=𝑃(𝐴∩𝐡).

Since we know 𝑃(𝐴), 𝑃(𝐡), and 𝑃(𝐴βˆͺ𝐡), but not 𝑃(𝐴∩𝐡) or 𝑃(𝐡), we need to use rules of probability for the complement of an event and the probability of the union of two events. First, we will use the rule of probability for a complement of an event to find 𝑃(𝐡).

We know that 𝑃(𝐡)=1βˆ’π‘ƒ(𝐡).

So, to find 𝑃(𝐡), we substitute 𝑃(𝐡)=0.47 and rearrange for 𝑃(𝐡): 0.47=1βˆ’π‘ƒ(𝐡)0.47+𝑃(𝐡)=1𝑃(𝐡)=1βˆ’0.47𝑃(𝐡)=0.53.

Having found 𝑃(𝐡), we can use the addition formula to find 𝑃(𝐴∩𝐡). This is 𝑃(𝐴βˆͺ𝐡)=𝑃(𝐴)+𝑃(𝐡)βˆ’π‘ƒ(𝐴∩𝐡).

By substituting 𝑃(𝐴)=0.71, 𝑃(𝐡)=0.53, and 𝑃(𝐴βˆͺ𝐡)=0.99 and then rearranging to make 𝑃(𝐴∩𝐡) the subject, we get 0.99=0.71+0.53βˆ’π‘ƒ(𝐴∩𝐡)0.99+𝑃(𝐴∩𝐡)=0.71+0.53𝑃(𝐴∩𝐡)=0.71+0.53βˆ’0.99=0.25.

Since we have found 𝑃(𝐴∩𝐡)=0.25, we can now use this, along with 𝑃(𝐡)=0.53, to find 𝑃(π΅βˆ’π΄). We do this by substituting into the formula and solving for 𝑃(π΅βˆ’π΄): 𝑃(π΅βˆ’π΄)=𝑃(𝐡)βˆ’π‘ƒ(𝐴∩𝐡)𝑃(π΅βˆ’π΄)=0.53βˆ’0.25=0.28.

Therefore, 𝑃(π΅βˆ’π΄)=0.28.

In this explainer, we have learned about the probability rule for the difference between two events 𝐴 and 𝐡. We have seen this applied to examples where only this rule is applied or where further rules, such as those for the complement of a set or the union of two sets, are used.

Key Points

  • The difference between two sets 𝐴 and 𝐡 is denoted by π΄βˆ’π΅ and is represented on the Venn diagram below:
  • The rule of probability for the difference between two events 𝐴 and 𝐡 is 𝑃(π΄βˆ’π΅)=𝑃(𝐴)βˆ’π‘ƒ(𝐴∩𝐡).
  • The rules of probability for the difference between two events, when 𝐡 is a subset of 𝐴, are
    • 𝑃(π΄βˆ’π΅)=𝑃(𝐴)βˆ’π‘ƒ(𝐡),
    • 𝑃(π΅βˆ’π΄)=0.

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