Generally there are three interpretations of probability 1 Classical

# Generally there are three interpretations of

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Generally, there are three interpretations of probability: 1 Classical probability; 2 Empirical or relative frequency probability; and 3 Subjective probability Classical interpretation of probability arose from games of chance. It uses sample space to determine numerical probability that an event will happen. The assumption is that all outcomes in the sample space are equally likely to occur. Don’t just assume that all outcomes are equally likely unless either you are told to, or there is some physical reason for assuming it! By Arthur Mpazi Yambayamba Introduction to Probability and Probability Distributions September 1, 2015 5 / 20
Classical Probability If there are n equally likely possibilities, of which one must occur, and m of these are regarded as favourable to an event, or as “success”, then the probability of the event or a “success”is given by m n . When all outcomes are equally likely: 1 count the number of outcomes in the sample space; say this is n . 2 count the number of outcomes in the event of interest, A , and say this is m . 3 P ( A ) = m n . Example 1 : A balanced die (with all outcomes equally likely) is rolled. Let A be the event that an even number occurs. There are six elements in the sample space, S = { 1 , 2 , 3 , 4 , 5 , 6 } , and there are three favourable outcomes (2, 4, 6) in A . Hence, P ( A ) = 3 6 = 1 2 . By Arthur Mpazi Yambayamba Introduction to Probability and Probability Distributions September 1, 2015 6 / 20
Classical Probability...Cont’d Example 2 : Suppose we toss two coins. Assume that the outcomes are equally likely (fair coin). (a) What is the sample space? (b) Let A be the event that at least one of the coins shows up heads. Find P ( A ). (c) What will be the sample space if we know that at least one of the coins showed up heads? (a) The sample space consists of four outcomes, namely S = { ( H , H ) , ( H , T ) , ( T , H ) , ( T , T ) } . (b) The event A has three outcomes, (H, H), (H, T), and (T, H). Therefore, P ( A ) = 3 4 . (c) Since we know that at least one of the coins showed up heads, the possible outcomes are (H, H), (H, T), and (T, H). The sample space now has only three outcomes, S = { ( H , H ) , ( H , T ) , ( T , H ) } . By Arthur Mpazi Yambayamba Introduction to Probability and Probability Distributions September 1, 2015 7 / 20
Empirical or Relative Frequency Probability The probability of an outcome (event) is the proportion of times the outcome (event) would occur in a long run of repeated experiments. Empirical probability relies on actual experience to determine the likelihood of outcomes. In empirical probability, for example, one might actually roll a given die or toss a coin 5,000 times, observe the various frequencies, and use these frequencies to determine the probability of an outcome! Symbolically, if an experiment is conducted n different times and if event A occurs on n A of these trials, then the probability of event A is approximately: P ( A ) n A n .

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• Fall '18
• F. TAILOKA
• Probability, Probability theory, Probability interpretations, Frequency probability, Arthur Mpazi Yambayamba