Addition Rules of Probability

Introduction

The Addition rules of probability determine the possibility of two simultaneous occurrences. Probability is a field of mathematics that measures the certainty or uncertainty of an event or series of occurrences. The Addition rules of probability specify two formulae, one for the likelihood of either of two mutually exclusive events occurring and the other for the probability of two non-mutually exclusive events occurring.

Keep reading these study material notes on Addition rules of probability to learn in detail about the topic.

Concepts Related to Addition Rules of Probability

It is necessary to grasp a few basic concepts to understand the Addition rules of probability. 

• Sample space: It’s the collection of all potential outcomes. For example, when flipping a coin, the sample space is Heads or Tails since all possible outcomes are heads and tails.

• Event: An event is a specific result in probability. For instance, tossing a coin and obtaining heads is an event.
• Mutually exclusive events: These events do not occur simultaneously. In the coin example, you cannot obtain tails if you get heads. 
• Non-mutually exclusive events: These events can occur simultaneously. For instance, driving while listening to the radio, even and prime numbers on a die, losing and scoring in a game, and exercising while sweating. 
• Independent events: These events can occur independently from one another. When flipping two coins, for example, the outcome of the second coin is unconnected or independent of the outcome of the first.

Following is the formula for calculating the probability of two occurrences A and B:

P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

P(A ∪ B) – Probability that either A or B happens

P(A) – Probability of Event A

P(B) – Probability of Event B

P(A ∩ B) – Probability of A and B happening together

Mutually Exclusive Event

Rolling the dice is an example of a mutually exclusive occurrence. Since the dice cannot fall on two sides at the same time, the possibility of each side is mutually exclusive. Suppose A and B are mutually exclusive events, P(AB) = 0. The likelihood that one of the events will occur is: P(A) or B) = P(A) + P(B). Therefore, the formula for calculating mutually exclusive events is as follows:

P(A ∪ B) = P(A) + P(B)

Some examples of mutually exclusive events:

1. Drawing red cards and clubs in a pack of 52 cards is mutually exclusive since all the clubs in a card are black. 
2. The events 4 and 6 in six-sided rolling dice are mutually exclusive since both events 4 and 5 cannot occur concurrently if we toss a single dice.

Follow these steps to determine the probability of mutually exclusive events:

1. Determine the total number of potential outcomes.
2. Determine the desired results.
3. For each event, create a ratio.
4. Add each event’s ratios or fractions.

Example:

Question: What is the probability of a dice showing number 1 or 6?

Solution: 

Let,

P(1) is the probability of getting a number 1

P(6) is the probability of getting a number 6

P(1) = 1/6 and P(6) = 1/6

So, P(1 or 6) = P(3) + P(5)

P(1 ∪ 6) = (1/6) + (1/6) = 2/6

P(1 ∪ 6) = 1/3

Therefore, the probability of a dice showing 1 or 6 is 1/3.

Non-mutually Exclusive Event

Non-mutual occurrences include selecting a black or seven card from a deck of standard playing cards. The intersection of two events is what you get when you seek for the possibility of two occurrences occurring at the same moment. The formula for calculating non-mutually exclusive event is as follows: 

P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

Follow these procedures to calculate the possibility of non-mutually exclusive events.

1. Calculate the sum of all possible results.
2. Determine the desired results.
3. Calculate a ratio for each occurrence.
4. Add each event’s ratios or fractions.
5. Subtract the overlap of the two events.

Example:

Question: What is the possibility that a single card drawn from a standard deck of cards will be an ace or a diamond?

Solution: 

Let X represent the event of drawing an ace and Y represent the event of picking a diamond.

P (X) = 4/52

P (Y) = 13/52

Since there is one positive scenario in which the card can be both an ace and a diamond, the two situations are not mutually exclusive.

P (X ∩ Y) = 1/52

P (X ∪ Y) = 4/52 + 13/52 − 1/52   =16/52 

P (X ∪ Y) = 4/13

Conclusion

The Addition rules of probability comprise two rules or formulae, one for two mutually exclusive occurrences and another for two non-mutually exclusive events. The first formula is the sum of the two occurrences’ probability. The second formula is the sum of the probabilities of the two events minus the likelihood of both occurring. Non-mutually exclusive indicates that there is some overlap between the two occurrences under consideration, and the formula adjusts for this by subtracting the probability of the overlap from the sum of the probabilities of Y and Z. The first form of the rule is a particular case of the second form, according to theory.