When it comes to calculating probabilities, there are two fundamental rules that we can use: the addition rule and the multiplication rule. These rules are essential when dealing with complex probability problems, especially when events are dependent or independent of each other. In this article, we will explore these rules in detail and understand how we can use them to calculate probabilities in real-world scenarios.

The addition rule is used to calculate the probability of two or more events occurring together. To understand this rule, let's consider an example. Suppose there are two bags of marbles. The first bag has ten red marbles and four blue marbles, while the second bag has five red marbles and six blue marbles. If we randomly select one marble from each bag, what is the probability of selecting two red marbles?

To solve this problem, we need to consider the two events separately. The first event is selecting a red marble from the first bag, and the probability of this is 10/14 since there are ten red marbles out of 14 marbles in total. The second event is selecting a red marble from the second bag, and the probability of this is 5/11. Using the addition rule, we can calculate the probability of both events occurring together as:

Probability of selecting two red marbles = Probability of selecting a red marble from the first bag + Probability of selecting a red marble from the second bag

= 10/14 x 5/11

= 0.267

Therefore, the probability of selecting two red marbles is 0.267 or approximately 27%.

The addition rule also applies to mutually exclusive events, which are events that cannot occur at the same time. For example, if we roll a dice, the probability of getting a 4 and the probability of getting a 5 are mutually exclusive since we cannot get both at the same time. In this case, the probability of either event occurring is the sum of their individual probabilities. For instance, the probability of rolling either a 4 or a 5 is:

Probability of rolling a 4 or a 5 = Probability of rolling a 4 + Probability of rolling a 5

= 1/6 + 1/6

= 1/3

Therefore, the probability of rolling either a 4 or a 5 is 1/3 or approximately 33%.

The multiplication rule is used to calculate the probability of two or more events occurring in a specific order. To understand this rule, let's consider an example. Suppose we have a box with three red balls and two blue balls. If we randomly select two balls without replacement, what is the probability of selecting a red ball first and a blue ball second?

To solve this problem, we need to consider the two events in the given order. The first event is selecting a red ball, and the probability of this is 3/5 since there are three red balls out of five balls in total. The second event is selecting a blue ball given that we have already selected a red ball, and the probability of this is 2/4 since there are only two blue balls left out of four balls in total. Using the multiplication rule, we can calculate the probability of both events occurring in the given order as:

Probability of selecting a red ball first and a blue ball second = Probability of selecting a red ball × Probability of selecting a blue ball given that we have already selected a red ball

= 3/5 x 2/4

= 0.3

Therefore, the probability of selecting a red ball first and a blue ball second is 0.3 or 30%.

The multiplication rule also applies to independent events, which are events where the outcome of one event does not affect the outcome of the other event. For example, if we roll a dice twice, the probability of getting a 4 and the probability of getting a 5 on the second roll are independent events since the outcome of the first roll does not affect the outcome of the second roll. In this case, the probability of both events occurring is the product of their individual probabilities. For instance, the probability of rolling a 4 on the first roll and a 5 on the second roll is:

Probability of rolling a 4 on the first roll and a 5 on the second roll = Probability of rolling a 4 on the first roll × Probability of rolling a 5 on the second roll

= 1/6 x 1/6

= 0.027

Therefore, the probability of rolling a 4 on the first roll and a 5 on the second roll is 0.027 or approximately 2.7%.

Often, we need to use both the addition rule and the multiplication rule to solve complex probability problems. For example, suppose we have a box with five red balls and four blue balls. We randomly select two balls and replace them after each selection. If we repeat this process three times, what is the probability of selecting two red balls and one blue ball?

To solve this problem, we need to break it down into smaller, manageable steps. First, we need to calculate the probability of selecting two red balls and one blue ball on one selection using the multiplication rule. The probability of selecting a red ball is 5/9, and the probability of selecting a blue ball is 4/9. Therefore, the probability of selecting two red balls and one blue ball in any order is:

Probability of selecting two red balls and one blue ball = Probability of selecting two red balls × Probability of selecting one blue ball

= (5/9)² × (4/9)

= 0.123

Therefore, the probability of selecting two red balls and one blue ball on one selection is 0.123 or approximately 12.3%.

Next, we need to use the addition rule to calculate the probability of selecting two red balls and one blue ball in any of the three selections. There are three ways this can happen, as follows:

- Select two red balls and then one blue ball
- Select one red ball, then one blue ball, and then one red ball
- Select one blue ball, then one red ball, and then one red ball

Using the addition rule, we can calculate the probability of any of these events occurring as:

Probability of selecting two red balls and one blue ball in any order = Probability of selecting two red balls and one blue ball on one selection + Probability of selecting one red ball, one blue ball, and one red ball on one selection + Probability of selecting one blue ball, one red ball, and one red ball on one selection

= 0.123 + (5/9) × (4/9) × (5/9) + (4/9) × (5/9) × (5/9)

= 0.344

Therefore, the probability of selecting two red balls and one blue ball over three selections is 0.344 or approximately 34.4%.

Combining probabilities using the addition rule and the multiplication rule is an essential skill in probability theory. These rules are used to calculate the probability of two or more events occurring together or in a specific order, depending on whether the events are dependent or independent. To solve complex probability problems, we often need to use both rules in combination. Understanding these rules can help us make informed decisions in real-world situations where probability is involved.