Probability 6

Probability Rules   Addition  Law: 1. Addition Law For Not Mutually exclusive Events when two or more events will happen at the same time, and the events are not mutually exclusive, then: P(X or Y) = P(X) + P(Y) - P(X Y) For example, what is the probability that a card chosen at random from a deck of cards will either be a king or a heart? P(King or Heart) = P(X or Y) = 4/52 + 13/52 - 1/52 = 30.77% Addition Law For  Mutually exclusive Events   when two or more events will happen at the same time, and the events are mutually exclusive, then: P(X or Y) = P(X) + P(Y) For example, suppose we have a machine that inserts a mixture of beans, broccoli, and other types of vegetables into a plastic bag. Most of the bags contain the correct weight, but because of slight variation in the size of the beans and other vegetables, a package might be slightly underweight or overweight. A check of many packages in the past indicate that: Weight.................Event............No. of Packages.........Probability Underweight..........X.......................100...........................0.025 Correct weight.......Y.......................3600.........................0.9 Overweight............Z.......................300...........................0.075 Total................................................4000......................1.00 What is the probability of selecting a package at random and having the package be under weight or over weight? Since the events are mutually exclusive, a package cannot be underweight and overweight at the same time. The answer is: P(X or Z) = P(0.025 + 0.075) = 0.1 B The Multiplication Law: 3  Multiplication Law For dependent Events when two or more events will happen at the same time, and the events are dependent, then the general rule of multiplication law is used to find the joint probability: P(X and Y) = P(X) . P(Y|X) For example, supposes there are 10 marbles in a bag, and 3 are defective. Two marbles are to be selected, one after the other without replacement. What is the probability of selecting a defective marble followed by another defective marble? Probability that the first marble selected is defective: P(X)=3/10 Probability that the second marble selected is defective: P(Y)=2/9 P(X and Y) = (3/10) . (2/9) = 7% This means that if this experiment were repeated 100 times, in the long run 7 experiments would result in defective marbles on both the first and second selections. Another example is selecting one card at random from a deck of cards and finding the probability that the card is an 8 and a diamond. P(8 and diamond) = (4/52) . (1/4) = 1/52 which is = P(diamond and 8) = (13/52) . (1/13) = 1/52. 4 Multiplication law For Independent events when two or more events will happen at the same time, and the events are independent, then the special rule of multiplication law is used to find the joint probability: P(X and Y) = P(X) . P(Y) if two coins are tossed, what is the probability of getting a tail on the first coin and a tail on the second coin? P(T and T) = (1/2) . (1/2) = 1/4 = 25%. This can be shown by listing all of the possible outcomes: T T, or T H, or H T, or H H. Games of chance in casinos, such as roulette and craps, consist of independent events. The next occurrence on the die or wheel should have nothing to do with what has already happened.