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This important characteristic of probability experiments is the known as the Law of Large Numbers : as the number of repetitions of an experiment is increased, the relative frequency obtained in the experiment tends to become closer and closer to the theoretical probability. Even though the outcomes don't happen according to any set pattern or order, overall, the long-term observed relative frequency will approach the theoretical probability. (The word empirical is often used instead of the word observed.) The Law of Large Numbers will be discussed again in Chapter 7.
It is important to realize that in many situations, the outcomes are not equally likely. A coin or die may be unfair , or biased . Two math professors in Europe had their statistics students test the Belgian 1 Euro coin and discovered that in 250 trials, a head was obtained 56% of the time and a tail was obtained 44% of the time. The data seem to show that the coin is not a fair coin; more repetitions would be helpful to draw a more accurate conclusion about such bias. Some dice may be biased. Look at the dice in a game you have at home; the spots on each face are usually small holes carved out and then painted to make the spots visible. Your dice may or may not be biased; it is possible that the outcomes may be affected by the slight weight differences due to the different numbers of holes in the faces. Gambling casinos have a lot of money depending on outcomes from rolling dice, so casino dice are made differently to eliminate bias. Casino dice have flat faces; the holes are completely filled with paint having the same density as the material that the dice are made out of so that each face is equally likely to occur. Later in this chapter we will learn techniques to use to work with probabilities for events that are not equally likely.
The complement of event $A$ is denoted $\mathrm{A\text{'}}$ (read "A prime"). $\mathrm{A\text{'}}$ consists of all outcomes that are NOT in $A$ . Notice that $\mathrm{P(A)\; +\; P(A\text{'})\; =\; 1}$ . For example, let $\mathrm{S\; =\; \{1,\; 2,\; 3,\; 4,\; 5,\; 6\}}$ and let $\mathrm{A\; =\; \{1,\; 2,\; 3,\; 4\}}$ . Then, $\mathrm{A\text{'}\; =\; \{5,\; 6\}.\; P(A)\; =}\frac{4}{6}\mathrm{,\; P(A\text{'})\; =}\frac{2}{6}\mathrm{,\; and\; P(A)\; +\; P(A\text{'})\; =}\frac{4}{6}+\frac{2}{6}\mathrm{=\; 1}$
The conditional probability of $A$ given $B$ is written $\mathrm{P(A|B)}$ . $\mathrm{P(A|B)}$ is the probability that event $A$ will occur given that the event $B$ has already occurred. A conditional reduces the sample space . We calculate the probability of $A$ from the reduced sample space $B$ . The formula to calculate $\mathrm{P(A|B)}$ is
$\mathrm{P(A|B)=}$ $\frac{\mathrm{P(A\; AND\; B)}}{\mathrm{P(B)}}\text{}$
where $\mathrm{P(B)}$ is greater than 0.
For example, suppose we toss one fair, six-sided die. The sample space $\mathrm{S\; =\; \{1,\; 2,\; 3,\; 4,\; 5,\; 6\}}$ . Let $A$ = face is 2 or 3 and $B$ = face is even (2, 4, 6). To calculate $\mathrm{P(A|B)}$ , we count the number of outcomes 2 or 3 in the sample space $\mathrm{B\; =\; \{2,\; 4,\; 6\}}$ . Then we divide that by the number of outcomes in $B$ (and not $S$ ).
We get the same result by using the formula. Remember that $S$ has 6 outcomes.
$\mathrm{P(A|B)}=$ $\frac{\text{P(A and B)}}{\text{P(B)}}\text{}=\frac{\text{(the number of outcomes that are 2 or 3 and even in S) / 6}}{\text{(the number of outcomes that are even in S) / 6}}=\frac{\mathrm{1/6}}{\mathrm{3/6}}=\frac{1}{3}$
In a particular college class, there are male and female students. Some students have long hair and some students have short hair.Write the symbols for the probabilities of the events for parts (a) through (j) below. (Note that you can't find numerical answers here. You were not given enough information to find any probability values yet; concentrate on understanding the symbols.)
**With contributions from Roberta Bloom
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