# bayes' theorem examples and solutions

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Bayes’ Theorem governs the likelihood that one event is based on the occurrence of some other events. [By Total Probability Theorem]. Find the probability that the transferred ball is black. The ball so drawn is found to be red in colour. If the two events are mutually exclusive, the probability of the union of the two events is the probability of the first event plus the probability of the second event. Of their outputs, 5,4 and 2 percent are respectively defective bolts. A screening test accurately detects the disease for 90% if people with The experiment is done and we  know that event X has occurred. Find out the probability that really an odd number appears. Let us denote the probability of throwing 6 on an unbiased dice be represented by P(D); So, the probability of throwing another number on the dice is represented by P(E) = 5/6, The probability of picking up a red card=and it is denoted by P(R1) = 1/2, So, the probability of not picking up a red card = 1 – ½ = ½, So, the probability of picking up a red card after drawing a 6 on the dice. This theorem enables us to evaluate P(Bi/X) if all the P(Bi) priori probabilities and P(X/Bi) likelihood probabilities are known. If the bolt drawn is found to be defective, what is the probability that it is manufactured by machine B. It depends upon the concepts of conditional probability. problem and check your answer with the step-by-step explanations. You tell a lie 4 out of 5 times. We have to find out the value of the probability of drawing a red ball from the second bag. Try the free Mathway calculator and Let E be an event associated with X, then by Bayes’ Theorem we get: According to conditional probability, we get, = P(Bi)P(X∣Bi)∑j=1nP(Bj)P(X∣Bj)\frac{P(B_i)P(X|B_i)}{\sum_{j=1}^n P(B_j)P(X|B_j)}∑j=1n​P(Bj​)P(X∣Bj​)P(Bi​)P(X∣Bi​)​ All of the machines can produce 1000 pins at a time. Let us take the probability of choosing a faulty pin randomly be represented by P(A); Pin choose from the first machine be represented by M1; Pin choose from the second machine be represented by M2: Pin choose from the third machine be represented by M3: Chance of choosing pin any one of the three machines = P(M1) = P(M2) = P(M3) = 1/3, Probability of choosing a faulty pin from 1st machine is. problem solver below to practice various math topics. You have been given dice and a pack of 52 cards. A biased coin (with probability of obtaining a Head equal to p > 0) is tossed repeatedly and independently until the ﬁrst head is observed. Let B1 be the event that the bolt is manufactured by machine A. B2 be the event that the bolt is manufactured by machine B. B3 be the event that the bolt is manufactured by machine C. X be the event that the bolt is defective. We know that the likelihood of heart disease increases with increasing age. The following diagram shows the Addition Rules for Probability: Mutually Exclusive Events and Non-Mutually Exclusive Events. You have to throw a dice and then you have to pick up a card. We have to find P(AA∪B),P\left( \frac{A}{A\cup B} \right),P(A∪BA​), which is equal to P[A∩(A∪B)]P(A∪B)=P(A)P(A∪B)=0.250.30=56P\frac{[A\cap (A\cup B)]}{P(A\cup B)}=\frac{P(A)}{P(A\cup B)}=\frac{0.25}{0.30}=\frac{5}{6}PP(A∪B)[A∩(A∪B)]​=P(A∪B)P(A)​=0.300.25​=65​. P(X/B1) = Probability that the bolt drawn is defective so that it is manufactured by machine A = 5/100, P(X/B2) = Probability that the bolt drawn is defective so that it is manufactured by machine B = 4/100, P(X/B3) = Probability that the bolt drawn is defective so that it is manufactured by machine C = 2/100, Therefore the probability that the bolt is manufactured by machine B given that the bolt drawn is defective = P(B2/X), = P(B2) P(X/B2) ÷ [P(B1)P(X/B1) +P(B2)P(X/B2)+P(B3)P(X/B3)], = (35/100)(4/100) ÷ [(25/100)(5/100)+(35/100)(4/100)+(40/100)(2/100)]. It is reported that an odd number on the dice appears. The following diagram shows the Multiplication Rules for Probability (Independent and Dependent Events) and Bayes' Theorem. What is the probability that it faces 4? If the bolt drawn is found to be defective, what is the probability that it is manufactured by machine B. be the event that the bolt is manufactured by machine A. Imagine you are a financial analyst at an investment bank. This theorem gives us the probability of some events depending on some conditions related to the event. Use of Bayes' Thereom Examples with Detailed Solutions Example 1 below is designed to explain the use of Bayes' theorem and also to interpret the results given by the theorem. Example 6: Find the  condition for E and F to be independent events such that 0