Multiplication Rule of Probability

Introduction

• Probability is the possibility of happening an event. In the other words, it is the ratio of the total number of favorable outcomes to the total number of favorable outcomes and if the probability is one it means an event is a sure event or if the probability is zero then it means that event will not happen.

• Probability is simply a useful description (in the form of a mathematical model) for experiments whose exact outcome is difficult to predict in advance

• When you toss a coin, it's tough to know in advance if a head or a tail will appear. When you can't predict the exact outcome, it's often useful to try to characterize every outcome that could occur along with a numerical description as to which are the most likely to occur.

• The numerical description you choose can be based on your experience, knowledge of physics, what makes for easy calculations, or many other factors.

• The usual model for the coin toss is to say that heads and tails are possible.

In this tutorial, we will discuss the multiplication rule of probability.

Probability

• Probability is the possibility of happening an event. In the other words, it is the ratio of the total number of favorable outcomes to the total number of favorable outcomes and if the probability is one it means an event is a sure event or if the probability is zero then it means that event will not happen.

  $$\mathrm{Probability\: (event) =\frac{favorable\: outcomes}{total\: outcomes}}$$

• Probability is the percentage of success.

• Probability is used to describe an outcome function for fixed parameter values.

• For example, if you toss a coin 10 times and it is a fair coin, what is the probability that it comes up heads each time? This can be calculated by the above formula.

Dependent and Independent Events

Independent events are events that occur independently of other events.

Multiplication rule of probability for independent events

$$\mathrm{ P(A ∩B) = P ( A ) P ( B ).}$$

Dependent events are events whose probabilities do affect one another.

The multiplication rule of probability for dependent events

$$\mathrm{P (A ∩ B) = P(A ) P ( B | A ).}$$

Conditional Probability

• In simple words, given that an event occurred, what is the probability that another event occurs?

• I will try to explain with an example, no numbers just plain English. Say on a fine morning you wanted to go out, you have two options −

• take a cab or drive to the place. Let us call the event driving their event 1.

• Your cousin decided to join you. Let's call the event of your cousin joining you event 2.

• Given that event 2 occurs, what is the probability that event 1 occurs?

• This is called conditional probability. Given that your cousin joins you, what is the probability that you drive there? The event of you driving to the venue is conditional on the event of your cousin joining you.

• There's another conditional here −

• The event of your cousin joining you given that you decided to drive to the venue. This is found using Bayes' Theorem.

Mathematically, assume that either Event A or Event B can occur. P (B / A) and read as "the probability of B given A".

This can be written out as −

$$\mathrm{P (A/ B)= \frac{P(B/A) P(A)}{P(B)}}$$

• This can be viewed geometrically by drawing out the relevant Venn diagrams. Put in words, knowing that A has occurred, the only way for B to occur is to fall into the intersection of the two events.

• You divide out the total probability of A occurring since it is no longer possible for the entire event A to take place.

Multiplication Rule of Probability

• Probability multiplication defines the condition between two specific events. For two events A and B associated with sample space S, A ∩ B indicates the event in which both events occurred.

• The general law of multiplying probabilities can be easily obtained by multiplying both sides of the conditional probability equation by the denominator.

Multiplication rule of probability for independent events

$$\mathrm{ P(A ∩B) = P ( A ) P ( B ). }$$

The multiplication rule of probability for dependent events

$$\mathrm{P (A ∩ B) = P(A ) P ( B | A ). }$$

Total Probability Theorem

The total probability theorem is the basis of the Bayes theorem. This theorem helps to find the probability that an event will occur in different partitions of the sample space. If A is an event that can occur in conjunction with any one of

B_1,B_2,B_3,......... that are mutually exclusive, then

$$\mathrm{P(A)=P(A∩B_1)+P (A∩B_2)+P( A∩B_3 )+P (A∩B_4)+\dotso\dotso= P (B_1).P (A /B_1) + P(B_2).P(A / B_2) + P ( B_3 ).P (A / B_3) + P (B_4).P(A / B_4) + \dotso\dotso}$$

The above representation of P(A) is called the total probability theorem.

The above representation of P(A) is called the total probability theorem.

Bayes Theorem

Bayes' theorem is a probability and statistical theorem named after Rev. Thomas Bayes and helps determine the probability of an event based on an event that has already occurred.

$$\mathrm{Formula : P (A/ B)= \frac{P(B/A) P(A)}{P(B)}}$$

Here, A and B are the two given events, and P (A/ B)is the probability of happening of event A if event B already happened.

Solved Examples

Example 1: Find the probability of getting 2 on a fair die?

Solution: Let A denote the event “getting 2.”

According to the question, sample space will be {1, 2, 3, 4, 5, 6}.

$$\mathrm{P(A)=\frac{1}{6}}$$

Example 2: What is the probability of getting 2, given you’ve rolled an even number?

Solution: You’ll note that the sample space has been now reduced to S′={2,4,6}. Assuming B to be the event “rolling an even number,” the probability is now:

$$\mathrm{P(A\: given\: B)=\frac{1}{3}}$$

Example 3: What is P (B | A) if P(A) = 0?

Solution: $\mathrm{P(B | A)=\frac{P ( A ∩ B )}{P ( A )}}$

$$\mathrm{But\: P(A )=0; (given)}$$

$$\mathrm{and\: range\: for\: any\: probability\: is\: (0 < P <1);}$$

$$\mathrm{Thus\: no\: matter\: what\: is\: P(B),}$$

$$\mathrm{P(A∩B)=0;}$$

Thus the solution of P(B|A) is undefined

Example 4: calculate the conditional probability P(A|B). We know that P(B) = 0.4 and P(AnB) = 0.1

Solution: Using the conditional probability formula, we get −

$$\mathrm{P(A | B) = P(A∩B) / P(B)}$$

$$\mathrm{P(A | B) = 0.1 / 0.4}$$

$$\mathrm{P(A | B) = 0.25}$$

Conclusion

Probability multiplication defines the state between two specific events. For two events A and B associated with sample space S, A ∩ B indicates the event in which both events occurred. This is also known as the probability theorem of multiplication.

Independent events are events that occur independently of other events and Bays Formula − $\mathrm{P (A/ B )=\frac{P(B/A)P (A)}{P(B)}}$

FAQs

1.What is the multiplication rule in probability?

Probability multiplication defines the state between two specific events. For two events A and B associated with sample space S, A∩B indicates the event in which both events occurred. This is also known as the probability theorem of multiplication.

2.How do you solve a conditional probability question?

The usual way to solve conditional probability questions is by applying the definition − $\mathrm{P (A/ B )=\frac{P(B/A)P (A)}{P(B)}}$

3. What is the Formula of probability?

Favorable outcomes divided by total outcomes.

$$\mathrm{Probability (event) =\frac{favorable\: outcomes}{total\: outcomes}}$$

4.What is dependent probability?

Dependent events are events whose probabilities do affect one another.

5.What is the range of probability?

The range of probability from 0 to 1