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The probability of an event is denoted as the ratio of favorable outcomes to the total number of outcomes. The multiplication rule of probability is a particular case of probability. It explains a condition between two events. Therefore, it is often termed conditional probability. It comes in handy when two events occur at the same time. It is also helpful to find probabilities of two occasions when they are both dependent and independent. Let’s learn more about multiplication rule of probability aling with solved examples.
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Probability was first mentioned accidentally on a commentary in 1477 by Dante. But its primary application started in the early 17th century by Pascal and Chevalier de Metre. They postulated various theorems on the occurrence of events and laid the foundation of probability. Finding the probability of an individual event in an experiment is straightforward.
Also Check: Types of events
What is the Multiplication Rule of Probability?
The multiplication rule of probability states that the probability of occurrence of both events X and Y are equal to the product of the probability of event Y occurring and the conditional probability that event X occurs when Y occurs.
If two events X and Y are dependent, then the probability of both events co-occurring is denoted by-
P (X ∩ Y) = P (Y) x P (X|Y)
If X and Y are two independent events in an experiment, then the probability of both events occurring simultaneously is given by:
P(X ∩ Y) = P(X) x P(Y)
Examples of dependent events:
- Going to jail after you have robbed a bank.
- Buying a flight ticket and boarding your flight.
- Not paying telephone bills when you did not make any calls.
If two events X and Y are independent, then the probability of occurrence of both events simultaneously is denoted by:
P (X ∩ Y) = P (Y) x P (X)
Examples of independent events:
- Getting a paycheck when you own a car.
- When you own a book and look for a café.
- Buying a coffee and then buying a pencil.
Multiplication Theorem of Probability Video Explanation
Multiplication Theorem on Probability Video Explanation
Also Check: Statistics
Proof
From the multiplication theorem of probability, we know that the conditional probability of occurrence of an event X after event Y has occurred is P (X|Y) and is given by-
P (X|Y) = P (X∩Y) / P (Y)
Where, P (Y) ≠ 0
P (X ∩ Y) = P (Y) × P (X|Y) ……………………………………. (1)
P (Y|X) = P (Y ∩ X) / P (X)
Where, P (X) ≠ 0.
P (Y ∩ X) = P (X) × P (Y|X)
Since, P (X ∩ Y) = P (Y ∩ X)
P (X ∩ Y) = P (X) × P (Y|X) ……………………………………… (2)
From (1) and (2), we get:
P (X ∩ Y) = P (Y) × P (X|Y) = P (X) × P (Y|X) where,
P (X) ≠ 0, P (Y) ≠ 0.
The above result is known as the multiplication rule of probability.
For two independent events X and Y, equation (2) can be modified as-
P (X ∩ Y) = P (Y) × P (X)
For more than two events X, Y, and Z, multiplication rule of probability is given by-
P (x ∩ Y ∩ Z) = P (X) x P (Y|X) x P (Z| (X ∩ Y)) = P (X) x P (Y|X) x P (Z|XY)
One can follow the same rule to find the probability of more than three events.
Also Check: Event
Multiplication Theorem of Probability
From the above multiplication rule of probability, we know
P (X ∩ Y) = P (X) × P (Y|X); if P (X) ≠ 0
P (X ∩ Y) = P (Y) × P (X|Y); if P (Y) ≠ 0
The multiplication theorem of probability states that if two independent events, X and Y, occur in a random experiment, the probability of simultaneous occurrence of two separate events will be equal to the product of their probabilities.
Therefore,
P (X ∩ Y) = P (X) x P (Y)
Also, we know from multiplication rule that P (X ∩ Y) = P (X) × P (Y|X)
Since X and Y are independent events, therefore,
P (Y|X) = P (Y)
We get, P (X ∩ Y) = P (X) x P(Y)
From this, we can conclude that the multiplication theorem is true.
Also check:
Multiplication Rule of Probability: Using Specific Rule
If two events X and Y are independent, only then specific rule of multiplication is valid. It is denoted by P (X and Y) =P (X) x P (Y). That means that this rule works only when the probability of X does not change its values when the value of Y changes, and vice versa. It is a very straightforward rule.
Example: If the probability of getting a red ball is 5/6 and the probability of getting a green ball is 5/7, then the probability of both the events happening at the same time is given by,
5/6 x 5/7 = 25/42.
Multiplication Rule of Probability: Using General Rule
One can use this rule for finding probabilities of both independent and non-independent events. One has to apply a little logic to the occurrence of events to see the final probability. Here are the two examples based on the general rule of multiplication of probability-
Example 1: - An urn contains 12 pink balls and 6 blue balls. Without replacement, two balls are drawn one after another. What is the probability that both balls drawn are pink?
Solution 1:
Let us assume, A and B be the probability of drawing the first ball pink and the second ball also pink, respectively. Then, we need to find the probability of P (A ∩ B).
The probability of drawing the first ball pink = 12/18
Since the replacement is not allowed, and the first ball has been drawn, the remaining pink balls in the urn are 17, out of which 11 are pink balls.
Therefore, probability of drawing other pink ball = 11/17
By multiplication rule, P (A ∩ B) = P (A) x P (A|B)
= 12/18 x 11/17 = 132/306.
Example 2: - A magician takes out two cards from a deck of cards, one after the other, without replacement. What is the probability of getting an ace of spade, and a card of heart, as first and second card, respectively?
Solution 2:
Since the cards are drawn without replacement, therefore, both the events are dependent upon each other.
As we know, there is only one ace of spades in a deck of cards. Therefore, the probability of drawing ace of spades = 1/52
Since the ace of spades has already been drawn and replacement is not allowed. Therefore, only 51 cards remain in the deck.
Out of these 51 cards, 13 are hearts. Therefore, the probability of drawing a card of heart = 13/51
From the multiplication rule,
Probability of drawing ace of spades and a heart of card = 1/52 x 13/51 = 1/204.
Things to Remember
- The multiplication rule of probability states that the probability of occurrence of both events X and Y are equal to the product of the probability of event Y occurring and the conditional probability that event X occurs when Y occurs.
- If two events X and Y are dependent, then the probability of both events co-occurring is denoted by- P (X ∩ Y) = P (Y) x P (X|Y)
- If two events X and Y are independent, only then specific rule of multiplication is valid. It is denoted by P (X and Y) =P (X) x P (Y).
- That means that this rule works only when the probability of X does not change its values when the value of Y changes, and vice versa. It is a very straightforward rule.
Also Check : Set Theory In Maths
Sample Questions
Ques: State and proof the theorem of total probability. (4 marks)
Ans: Let us assume,
{E1 ,E2 ,….,E3} is a partition of the sample space S, and suppose that each of the events E1 , E2 ,….,E3 as nonzero probability of occurrence.
And, let A be any event associated with S,
Then P(A) = P(E1) P(A/E1) + P(E2) P(A/E2) +...+P(En) P(A/En)}
Proof:
\(S = E_1 \cup E_2 ... \cup E_n \) and
\(E_i \cap E_j , i \neq j, \) i,j = 1,2,….n
\(A = A \cap S = A \cap (E_1 \cup E_2 \cup ...\cup E_n)\)
=\((A \cap E_1) \cup(A \cap E_2) \cup ....\cup (A \cap E_n)\)
Thus, P(A) = \(P((A \cap E_1) \cup(A \cap E_2) \cup....\cup(A \cap E_n))\)
= \(P(A \cap E_1) + P(A \cap E_2) + ... + P(A \cap E_n)\)
By multiplication rule of probability we have;
\(P(A) = P(E_1) P(\frac{A}{E_1}) + P(E_2) P(\frac{A}{E_2}) + ......\)
\(+ P(E_n)P(\frac{A}{E_n})\)
By the multiplication rule of probability, we have,
P(A) = P(E1) P(A/E1) + P(E2) P(A/E2) + … + P(En) P(A/En)
Ques: If a fair coin is tossed 10 times, find out the probability that the outcome is exactly 6 heads? (3 marks)
Ans: Let X be the random variable of number of heads in an experiment of 10 trials.
Hence, clearly X has a Binomial Distribution with n = 10
Here n = 10, p = 1/2, q = 1 – p = ½
\(=> P(X = x) =C_x[\frac{1}{2}]^{10 - x} [\frac{1}{2}]^x = 10C_x[\frac{1}{2}]^{10}\)
\(P( X = 6) =10C _6 [\frac{1}{2}]^{10} = 10C_4 [\frac{1}{2}]^{10}\)
\(= \frac{10.9.8.7}{1.2.3.4} \frac{1}{2^{10}} = \frac{105}{512}\)
Ques: What is the difference between probability and statistics? (2 marks)
Ans: Probability is the branch of mathematics that deals with predicting the likelihood of future events. On the other hand, statistics deals with the analysis of the frequency of past events. Thus, statistics is a primary branch of applied mathematics, which tries to make sense of observations in the real world.
Ques: What is conditional probability? (2 marks)
Ans: Conditional probability deals with finding the probabilities of events when one or more event has already occurred. For example, finding the probability of winning the lottery after buying the lottery comes under conditional probability.
Ques: Two cards are drawn at random without replacement from a pack of 52 playing cards. What is the probability that both the cards are black? (4 marks)
Ans: The description of the events are as follows:
B1: probability of getting a black card in the first draw.
B2: probability of getting a black card in the second draw.
P(B1) = 26/52 = ½
When the first event is performed and as no replacement is allowed, the remaining total number of cards become 51 and black cards become 25.
P (B2/B1) = 25/ 51
Thus, the required probability is, P(B1 ∩ B2) = P(Bl) × P(B2/B1) =
½ x 25/51 = 25/ 102
Ques: What is the multiplication law of probability? (2 marks)
Ans: According to the multiplication law of probability, if A and B are two independent events in a probability experiment, then the probability that both events co-occur is: P (A and B) = P (A) ⋅ P (B)
Ques: How do you use the addition rule for probability? (2 marks)
Ans: According to the addition rule of probability, if A and B are two events in a probability experiment, then the probability that either one of the events will occur is: P (A or B) = P (A) + P (B) – P (A and B)
Ques: What are the five rules of probability? (3 marks)
Ans: The five rules of probability are-
- Probability Rule One (For any event A, 0 ≤ P(A) ≤ 1)
- Probability Rule Two (The sum of the probabilities of all possible outcomes is 1)
- Probability Rule Three (The Complement Rule)
- Probabilities Involving Multiple Events.
- Probability Rule Four (Addition Rule for Disjoint Events)
Ques: What are the Bayes theorem and its application? (3 marks)
Ans: Thomas Bayes, an 18th-century mathematician, gave a mathematical formula for determining conditional probability. His statement provides a way to revise existing predictions or theories (update probabilities) given new or additional evidence. According to Bayes’ theorem, the total probability rule (also known as the law of total probability) is fundamental in statistics relating to the conditional and marginal of an event based on prior knowledge of the conditions that might be relevant to the event.
Ques: What is basic probability? (2 marks)
Ans: Basic probability is the probability of a number that reflects the chance or likelihood that a particular event will occur. Probabilities can be expressed as proportions that range from 0 to 1, and they can also be expressed as percentages ranging from 0% to 100%.
Ques: What is the first law of probability? (2 marks)
Ans: The First Law of Probability states that the results of one chance event do not affect the results of subsequent chance events. Thus, the probability of obtaining heads the second time you flip it remains at ½. For example, even if you obtained five heads in a row, the odds of heads resulting from a sixth flip remain at ½.
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