Probability - Probability Rules of probability Probability...

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Unformatted text preview: Probability Rules of probability Probability Probability is a measure of the likelihood that a particular event will occur in any one trial or experiment carried out under prescribed conditions. Each separate possible result is called an outcome. Types of Probability (a) Empirical or experimental probability is based on previous known results. The relative frequency of the number of times the event has previously occurred is taken as an indication of likely occurrences in the future. Empirical or experimental probability Example 1: A random sample batch of 240 components is subjected to strict inspection and 20 items are found to be defective. If any one component is picked at random, chance of it being faulty is ‘20 in 240’ or ‘1 in 12’. Empirical or experimental probability If A is the event “a faulty component”, then the probability of A happening, written as: P(A) = 1 in 12 or 1/12 or 0.0833. Probability Expectation E is defined as the product of the number of trials N and the probability of the event A occurring in any one trial. E = N.P(A) Expectation Therefore a run of 600 components from the same machine would be expected to contain 50 defectives: 1 = x 600 = x x = 50 12 600 12 This does not mean that there will definitely be 50 defectives only that 50 are expected.) Types of Probability (b) Classical probability is based on a consideration of the theoretical number of ways in which it is possible for an event to occur. Classical probability Example 2: The probability that an event A might occur is determined as follows: P(A) = The number of ways in which event A can occur divided by the total number of all possible outcomes Classical probability Consider an unbiased die. When it is rolled the total possible number of outcomes is 6. Therefore the probability of throwing a 6 is 1/6 P(6) =1/6, while the probability of throwing a 5 is also 1/6, P(5) =1/6. Classical probability If out of n possible outcomes of a trial, it is possible for an event A to occur in x ways, and for event A not to occur in y ways, then n = x+ y. And so P(A) = x = x n x+y A Classical probability If an event A is certain to happen every time then x= n, y = 0, P(A) = n = 1. n If an event A cannot possibly occur at any time, then x = 0, y = n, P ( A) = 0 = 0. n Probability Mutually exclusive events are events which cannot occur at the same time. Mutually non-exclusive events can occur simultaneously. Probability Success or Failure: Since we are concerned with the probability of the occurrence of a particular event, when it does occur it is recorded as a success, and when it does not, a failure is recorded. Probability The probability of a success i.e. that of say event A occurring is represented by P(A), while the probability of a failure, that of event A not occurring is represented by P ( A) Probability Also the sum of all possible probabilities is: P ( A) P ( A) 1 Independence Independent Events: When the occurrence of one event does not affect the probability of the occurrence of the second event, the two events are said to be independent. Independence Dependent events: When one event affects the probability of the occurrence of the second. A pack of cards is shuffled and one card is picked out, say an ace: P(A) = 4/52 = 1/13. If the card is not replaced after the first draw the probability of picking an ace on the second draw is now 3/51. Independence The probability of the second event is dependent on the outcome of the first event. This is denoted by P(B/A ), the probability of B occurring given that A has happened. Independence Example 3: What is the probability of drawing two aces from a pack of cards if the first ace is not replaced ? P(Ace) = 4/52 =1/13 = 0.0769, (No replacement). P(Second ace) = 3/51 = 0.0588 (correct to 4 decimal places). Probability Laws Multiplication Law: Example 4: Consider the probability of both events A and B where event A represents throwing a six when rolling a die, and event B represents picking an ace from a pack of cards. Probability Laws Roll both die and pick a card as one trial. There are 6 possible outcomes for the die and 52 possible outcomes from the cards giving (6.52) outcomes altogether. There are 4 possibilities of obtaining a 6 and ace together, so the probability of event A and event B occurring is: P(A and B) = 4 = 1 x 4 = 1 . 4 6 x 52 6 x 52 6 52 Probability Laws When A and B are independent events, P(A and B) = P(A).P(B). Probability Laws Addition Law: Example 5: Consider the probability of events A and B where event A now represents the probability of picking a 7 from a deck of cards, and event B represents picking an ace. Probability Laws If only one card is being picked then the result can only be either an ace or a 7, (these are mutually exclusive events). The probability of event A or event B occurring is: P(A or B) = 4 + 4 = 4 + 4 = 1 + 1 = 2 13 52 52 52 = P(A) + P(B). 13 13 Probability Laws When A and B are two mutually exclusive events: P(A or B) = P(A) +P(B). Probability Laws Example 6: Consider the probability of picking a red card or an ace. Let event A be the probability of picking a red card, while event B represents the probability of picking an ace. Both events could occur simultaneously (not mutually exclusive events). However we are only interested in the probability of either event A or event B not both. Probability Laws P(A or B) = 26 + 4 = 26 + 4 52 52 52 This follows the same trend as before but we must take into account the probability of a red ace which we do not want! Therefore P(A or B) = 26 + 4 - 2 = 28 = 7 52 52 52 52 = P(A) or P(B) - P(A and B). 13 Probability Laws When A and B are not mutually exclusive events, P(AorB) = P(A) + P(B) -P(A).P(B) Probability Laws Example7: Consider the probability of scoring a multiple of 3 i.e. (3 or 6) when a die is rolled. Let this be event A. P(A) = 1/6 +1/6 = 2/6. Now consider the probability of throwing a multiple of 2 i.e. (2, 4 or 6). Let this be event B. P(B) = 1/6 + 1/6 +1/6 = 3/6. Probability Laws Now what is the probability of scoring a multiple of 2 or 3? Again these events are not mutually exclusive since 6 is a multiple of both 2 and 3. Therefore the probability of scoring one of these sixes must be removed: Probability Laws P(A or B) = P(A) or P(B) - P(A and B) = 2/6 + 3/6 - (2/6.3/6) = 2/6 + 3/6 - 6/36 = 2/6 + 3/6 - 1/6 = 2/3 Probability Laws Summary: Multiplication Law: ‘AND’ means multiply. Addition Law: ‘EITHER /OR’ means add Mutually Exclusive: Cannot happen at same time ...
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