<text><span class="style10">robability: Chance and Choice (1 of 2)</span><span class="style7">Not all actions and happenings have completely predictable results. Often we know that there is only a limited range of possible outcomes, but we do not know with certainty which of these to expect.</span><span class="style26">Probability theory</span><span class="style7"> enables us to describe with mathematical rigor the chance of an action or happening having a particular outcome. We may not, even then, make the right choice, but it will at least be a </span><span class="style26">justifiable</span><span class="style7"> choice.</span><span class="style10">Probability and frequency</span><span class="style7">When we toss a coin or throw a die, we cannot predict which of the sides will land facing upwards - this, after all, is the point of tossing coins and throwing dice. Assuming we accept the fairness of the coin and the way it is tossed, we know it is just as likely to come up heads as tails, and there is no other possible outcome. Similarly, with a fair die, it is just as likely to fall with any of its numbers - from 1 to 6 - upwards, and there are no other possible outcomes. We describe these examples by saying that all the possible </span><span class="style26">outcomes</span><span class="style7"> are </span><span class="style26">equiprobable</span><span class="style7">, and that the </span><span class="style26">a priori probability</span><span class="style7"> (i.e. the theoretical probability) of a coin coming up heads is 1 in 2 or 1/2, and that of throwing a 6 on a single die is 1 in 6 or 1/6.On the other hand, </span><span class="style26">empirical</span><span class="style7"> </span><span class="style26">probability</span><span class="style7"> (often called </span><span class="style26">a posteriori probability</span><span class="style7">) is based on observation and experiment. Here, the probability of a particular outcome is calculated from the proportion of times it has been observed to have happened before under the same conditions - its </span><span class="style26">relative frequency</span><span class="style7">. Thus, if you tossed a coin 10 times and the coin came up heads 3 times, the empirical probability that one of these throws came up heads is 3/10.</span><span class="style10">The probability scale</span><span class="style7">When an outcome is certain, it occurs every time: 1 in 1, 2 in 2, etc. Expressing this as a fraction, we say the probability is 1/1, that is, one. When an outcome is impossible, it occurs no times in any number of tests, so we say the probability is zero. For example, when throwing a die, the probability of throwing a number greater than 6 is zero, and the probability of throwing a number between 1 and 6 is one.Probabilities between certainty and impossibility are expressed as fractions. So if, for example, we know that the 6 sides of a die are equiprobable, and the probability of throwing any of them is 1, the probability of each must be 1/6. Furthermore, if we consider only two possible outcomes, an odd number or an even, the probability of each must be 1/2. The fact that there are three odd outcomes each with probability 1/6, and 1/6 + 1/6 + 1/6 = 1/2, demonstrates, very simply, the </span><span class="style26">addition rule</span><span class="style7">: we can add up the individual probabilities of the different possible outcomes in a particular trial to get the combined probability.Particularly in gaming and gambling, we see </span><span class="style26">odds</span><span class="style7"> used as a scale for measuring chance. 'Odds' - more formally known as </span><span class="style26">likelihood ratio</span><span class="style7"> - means the proportion of favorable to unfavorable possibilities, and is a different way of expressing probability. As we have seen, the probability of throwing, say, a 4, with a die is 1/6. Therefore, the probability of not throwing a 4 is 5/6. The 'odds' are thus expressed as 1 to 5 on throwing a 4 (or 5 to 1 against throwing a 4).</span><span class="style10">The law of large numbers</span><span class="style7">Suppose we toss a coin 10 times and the outcome is only 3 heads. The probability of a head is 1/2, so why do we not get 5 heads? We try a total of 100 tosses of the coin and the outcome is now, say, 40 heads, the last 6 being all heads. A gambler might back the chance that the 101st toss would fall tails, because, previously, there had been more tails than heads. Another gambler might back heads, because there seemed to be a 'run of heads', which would conform with the so-called 'law of averages'.However, we know that the probability of a head or a tail at any one toss is 1/2 and a coin cannot remember - it cannot be influenced by what has gone before. Both gamblers are relying on empirical probability where theoretical probability is what matters - both are therefore betting on hope.There is no 'law of averages'. Experimental and theoretical probability are connected only by the </span><span class="style26">law of large numbers</span><span class="style7">, which states that as the number of trials increases, the observed empirical probability comes closer and closer to the theoretical value. Thus in this example it means only that in the </span><span class="style26">very</span><span class="style7"> long run, the relative frequency settles down towards 1/2.</span></text>
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<text><span class="style10">ard games</span><span class="style7"> offer very many more combinations of possibilities than tossing coins or throwing dice. Some of the odds are staggering: for example, the odds against dealing 13 cards of one suit are 158 753 389 899 to 1, while the odds against a named player receiving a 'perfect hand' consisting of all 13 spades are 653 013 559 599 to 1. The odds against each of four players receiving a complete suit (a 'perfect hand') are in excess of 2 x 1027 to 1. </span></text>
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<text>ΓÇó MATHEMATICS AND ITS APPLICATIONSΓÇó NUMBER SYSTEMS AND ALGEBRA</text>
