Multiplication Rule for Independent Events

There are many situations in which we are interested in finding the probability of two events occurring simultaneously. Some of them are:

• Find the probability of rolling a double six when rolling two dice simultaneously or one after the other.
• Find the probability that a person chosen at random from a group is both female and dark-skinned.
• The probability of choosing a pair of students of the opposite sex from a section of the school.
• The probability that two redundant control systems fail at the same time in a space rocket launch.

This class of problems can be solved by means of the general rule of multiplication of probabilities. This rule establishes that, for two events A and B, the probability that they occur simultaneously, that is, the probability of intersection, is given by:

In this equation, P(A|B) is the conditional probability that event A occurs given B. The above is the general multiplication rule and applies to any pair of events. In some cases, the conditional probability is unknown or difficult to determine; however, in the case of independent events, this probability is simplified to give rise to the multiplication rule for independent events.

Multiplication Rule for Independent Events

What are independent events?

Two events A and B are independent of each other if the occurrence of one of them does not affect the probability that the other will occur. In mathematical terms, this implies that the conditional probability of either event occurring, given that we know the other has occurred, is equal to the simple probability of the first occurring. In other words, two events will be independent only if:

The interpretation of the above is that the probability of A occurring given that B has occurred is equal to the probability of A occurring. This implies that the occurrence of B did not affect the probability of A occurring, so both events occur. in an independent way.

Any pair of events that do not satisfy the above condition will be dependent events.

How is the multiplication rule affected in this case?

As we can see, the first expression of the independence condition can be used to simplify the general multiplication rule, since the first factor can be replaced by the simple probability of A, thus obtaining the following expression:

The above expression is known as the rule of multiplication of probabilities for independent events . It implies that if we know that two events are independent of each other and we know their probabilities of occurrence, then we can find the probability that they will both occur at the same time simply by multiplying these probabilities.

Examples of Independent Events

Lack of information can make it difficult to identify whether two events are independent. For example, we might think that having brown hair has nothing to do with the occurrence of breast cancer, but the physiology of the human body is so complex that no doctor would dare to make that statement.

However, there are many simple experiments in which we can easily identify whether or not two events are independent.

• Throw two dice at the same time. When rolling two dice, the result of one does not affect in any way the result that may appear on the other, so the event that one die lands on a given number is independent of the event that the other die lands on another number. or the same, even.
• The results of rolling the same die twice in a row are also independent of each other for the same reasons.
• Flip a coin twice. The fact that it lands heads or tails the first time will not affect the outcome of the next toss.
• In a refrigerator factory that has two independent production lines for components that use separate raw materials and labor, it is acceptable to assume that the probability that one of the two components will fail is independent of the probability that the other will fail.
• Randomly drawing a card or deck from a deck, replacing it, and then randomly drawing another card from the deck are separate events, since replacing the original card in the deck resets the chances of drawing any of the original cards.

Examples of events that are not independent

• Randomly drawing a card or deck from a deck and then drawing another card from the same deck without replacing the first one are not independent events, since drawing the first one reduces the total number of cards present in the deck, which affects the probability of any other card coming out. Also, if we don’t replace the first card, the probability of that card coming out the second time becomes zero.
• In a running car, the probability that the car’s engine will overheat and the probability that the water pump that cools the engine will fail are not independent events, since if the water pump fails, it becomes much more likely. the engine overheats.
• An even easier example to understand is that getting good grades in statistics is not independent of studying , since if we study, we are more likely to get good grades.

Examples of probability calculations using the multiplication rule for independent events

Example 1: Tossing a coin twice

Suppose we want to calculate the probability that when tossing a coin twice, the result is heads on both tosses.

If we call A the event in which the first toss lands heads and B the event in which the second toss lands heads, then the probability that we are asked to calculate is the probability of intersection of A with B, since we want both events happen. That is, the unknown is P(A∩B).

Since there are only two possible outcomes for each toss, the probability of either event occurring is the same:

Now, since we know that the events are independent, we can use the multiplication rule to determine the probability of intersection:

Example 2: Throwing two dice

Let’s calculate the probability that, when rolling two common six-sided dice, one of them lands on one and the second lands on an even number.

Let’s call the following events A and B:

       A = one of the dice lands on 1.

       B = one of the dice lands on an even number.

What we want to calculate is, again, P(A∩B).

Since the result of each die is independent of the number that results in the other, we can calculate P(A∩B) using the multiplication rule for independent events. But first, we need the probabilities of A and B.

The die has 6 faces with the numbers from 1 to 6, which do not repeat. Therefore, there is only one 1, and there are three even numbers, namely 2, 4, and 6. Therefore, the probabilities of the separate events occurring are:

Using these probabilities and the multiplication rule, we obtain the desired probability:

Example 3: Parts that fail

A factory that builds computer equipment uses, among other components, two different chips or integrated circuits from two different manufacturers. According to the manufacturer of the first chip, the probability that it will fail under normal operating conditions is 0.00133. For its part, the second manufacturer boasts that only two of its chips fail for every 5,000 units installed. The factory owner wants to find the probability that both components will fail at the same time. The failure of each chip brand can be considered independent of the other.

In this case, the statement itself specifies that the two events are independent, so we can use the multiplication rule above. In addition, the probability of the first chip failing is also provided, which we will call event A. The probability of the second chip failing (event B) can be calculated from the information provided by the manufacturer:

So the probability that both components fail at the same time is:

References

Conditional Probability and Independence . (nd). University of Florida Health. https://bolt.mph.ufl.edu/6050-6052/unit-3/module-7/

Devore, JL (1998). PROBABILITY AND STATISTICS FOR ENGINEERING AND SCIENCES . International Thomson Publishers, SA

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Multiplication Rule (Probability) [Examples] . (nd). Fhybea. https://www.fhybea.com/multiplication-rule.html

The general multiplication rule . (nd). Khan Academy. https://www.khanacademy.org/math/ap-statistics/probability-ap/probability-multiplication-rule/a/general-multiplication-rule