Essentials of Data Science – Probability and Statistical Inference – Multiplication Theorem of Probability

By / February 9, 2022

In the previous note on probability and statistical inference, we have seen axioms of probability, counting principles, and conditional probability. In this note, we will learn and understand the multiplication theorem of probability.

For any two arbitrary event A and B, the following holds:

P(A|B) = \frac{P(A \cap B)}{P(B)}

The same we can write in the following ways:

P(A \cap B) = P(A|B) P(B) = P(B|A) P(A)

This theorem does not require that P(A) > 0 \text{ and } P(B) > 0.

A generalisation of this result provides an expression for the probability of the intersection of an arbitrary number of events. 

Table of Contents hide
1 Multiplication theorem of Probability
2 References
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Multiplication theorem of Probability

Let us assume that we have A_1, A_2, A_3, A_4, \cdots, A_m events. So the probability of intersection of all are as follows:

P(A_1 \cap A_2 \cap \cdots \cap A_m) = P(A_1)P(A_2|A_1) P(A_3| A_1 \cap A_2) \cdots P(A_m | A_1 \cap A_2 \cdots A_{m-2} \cap A_{m-1)}

This theorem is very useful, when we do Bayesian analysis. It is a method of statistical inference that allows one to combine prior information about a population parameter with evidence from information contained in a sample to guide the statistical inference process.

Example 1:

A student figures that there is a 30% chance that he will be selected in the cricket team. If it does, he has 60% certain that he will be selected as Captain of the team. What is the probability that the student will be the captain in the selected team? 

Solution:

Let T be an event that the student will be selected in the team and C be an event that the student will be made the captain. Then the desired probability is P(T \cap C) = P(T) P(C|T) = \frac{30}{100} \times \frac{60}{100} = 0.18.

Probability of selected in cricket team and as a captain
Probability of selected in the cricket team and as a captain

So there is an 18% chance (negligible chance) that the student will be the captain. We can easily guess from the tree diagram that chance must be less than 30% by seeing the diagram. As selection in the Cricket team is only 30% and not selection in Cricket team is 70%.

Example 2:

A student is undecided as to whether to take a French course or a chemistry course. He estimates that his probability of receiving an A grade would be 1/2 in a French course, and 2/3 in a chemistry course. If he decides to base his decision on the flop of a fair coin, what is the probability that he gets an A in chemistry?

Solution:

Let C be an event that student takes chemistry and A be an event that he receives an A in whatever course he takes, then the desire probability is P(C \cap A) = P(C) P(A | C) = \frac{1}{2} \times \frac{2}{3} = \frac{1}{3}.

Example 3:

An ordinary deck of 52 playing cards is randomly divided into 4 piles of 13 cards each. Compute the probability that each pile has exactly 1 ace. 

Solution:

Define events E_i, i = 1,2,3,4 as follows:

  • E_1 = Event that the ace of spades is in any one of the piles. 
  • E_2 = Event that the ace of spades and the ace of hearts are in different piles. 
  • E_3 = Event that the aces of spades, hearts, and diamods are all in different piles. 
  • E_4 = Event that all 4 aces are in differnt piles. 

The desire probability is P(E_1 \cap E_2 \cap E_3 \cap E_4) = P(E_1) P(E_2|E_1)P(E_3|E_1 \cap E_2)P(E_4|E_1 \cap E_2 \cap E_3)

  • P(E_1) = 1, since E_1 is the sample sampe \Omega.
  • P(E_2 | E_1) = \frac{39}{51}, since the pile containing the ace of spades will receive 12 of the remaining 51 cards.
  • P(E_2 | E_2 \cap E_1) = \frac{26}{50}, since the piles containing the aces of spades and hearts will receive 24 of the remaining 50 cards; and finally, P(E_4 | E_1 \cap E_2 \cap E_3) = \frac{13}{49}.

Therefore, we obtain that the probability that each pile has exactly 1 ace is: 

P(E_1 \cap E_2 \cap E_3 \cap E_4) = \frac{39}{51} \frac{26}{50} \frac{13}{49} = 0.105

There is approximately a 10.5 percent chance that each pile will contian an ace. 

References

  1. Essentials of Data Science With R Software – 1: Probability and Statistical Inference, By Prof. Shalabh, Dept. of Mathematics and Statistics, IIT Kanpur.

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