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David Terr
Ph.D. Math, UC Berkeley

 

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<< 10.6. Counting Principles: Permutations and Combinations

 

10.7. Probability

In the previous section we laid down the building blocks of probability theory, namely permutations and combinations. The study of these two concepts and other ways of counting combinations of objects is known as combinatorics, and this is the starting point for studying probability.

So what do we mean by probability? Probability is a measure of how likely something is to happen. We are all familiar with probability in one way or another. When a weather reporter says there is a 60% chance of rain tomorrow, they are making a statement about the probability that it will rain the following day. Gambling involves probability. When a gambler makes a bet, he is hoping to win money based on the probability that he will win the bet.

So how do we measure probability? Roughly speaking, probability is a measure of relative frequency. In other words, the probability of an event E is equal to the number of equally likely ways E can occur divided by the total number of equally likely things which can occur. A few examples should make this clear. Consider the roll of a fair die. A fair die has six sides, and each side is equally likely to turn up when rolled. So what is the probability that the die will land on 5? There is only one way this can happen, but there are six possible outcomes, namely any particular one of the six numbers turning up. Thus, the probability of rolling a 5 is 1/6.

As a slightly more complicated example, let us compute the probability of rolling a 7 for the sum of two dice. How many ways can this happen? Here we need to consider each die separately. The first die can land on any number from 1 to 6, and so can the second. Thus, there are a total of 62 = 36 possible equally-likely outcomes for the roll of two dice. Of these 36 outcomes, how many of them correspond to rolling 7? Let (a,b) represent a roll of a for the first die and b for the second die. Then we must have a+b = 7. The possible rolls for which this holds are (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1). Thus, we see that there are 6 possible rolls of 7 out of 36 possible rolls, so the probability of rolling 7 is 6/36 = 1/6.

 

Example 1: What is the probability of rolling 4 as the sum of two dice?

Solution: There are three ways this can happen, namely (1,3), (2,2), (3,1). Thus, the probability of rolling 4 is 3/36 = 1/12.

 

Example 2: Yahtzee is a popular game involving five dice. A Yahtzee is a roll of five equal dice, for instance, five twos or five fours. What is the probability of rolling a Yahtzee on the first roll?

Solution: There are six possible ways of rolling a Yahtzee: all ones, all twos, all threes, all fours, all fives, or all sixes. On the other hand, the total number of possible rolls is 65. Thus, the probability of rolling a Yahtzee on the first roll is 6/65 = 6-4 = 1/1296.

 

Example 3: A royal flush is the highest possible poker hand. It consists of an ace, king, queen, jack, and ten, all of the same suit (clubs, diamonds, hearts, or spades). What is the probability of being dealt a royal flush on the first deal in a poker game?

Solution: There are only four possible ways of being dealt a royal flush: all clubs, all diamonds, all hearts, or all spades. On the other hand, as we saw at the end of the last section, there are a total of 2,598,960 possible poker deals. Thus, the probability of being dealt a royal flush is 4 / 2,598,960 = 1 / 649,740, so the next time you play poker, don't expect to be dealt one!

 

So where does combinatorics fit into probability theory? Combinatorics figures in when we consider an event consisting of repeated trials, for instance, tossing a fair coin many times. Suppose we toss a fair coin 10 times. We know the probability of the coin landing on heads for each toss is 1/2, since there are two possible equally-likely outcomes (heads or tails) and just one way it can come up heads. Furthermore, each toss of the coin is independent of every other toss. This basically means that the coin has no memory; the probability of the coin landing on heads on any given toss is always 1/2, regardless of the history of the previous tosses. As obvious as this may seem, gamblers often assume otherwise. Suppose the coin had just landed on heads 8 times in a row. Would you think the coin is more likely to land on tails on the next toss? You might think so, since you expect the percentage of heads to be close to 50%. But the probability of it landing on heads is always 1/2, since the coin has no memory. It is true that in the long run, 50% of the tosses will be heads and 50% tails, but this is not achieved by the coin making up for any defecit of heads or tails but rather by turning up heads roughly half the time in all future tosses.

So let us return to our example of tossing a fair coin 10 times. What is the probability of getting exactly 7 heads? This probability is equal to the number of possible ways of getting 7 heads divided by the total number of possible outcomes of 10 tosses. But the number of possible ways of getting 7 heads out of 10 tosses is equal to 10C7, since this is equal to the number of possible ways of choosing which 7 out of 10 tosses are heads. On the other hand, the total number of possible outcomes of 10 tosses is 210, since there are two possible outcomes for each toss and 10 tosses. A typical outcome might be HTTTHTHHTT, for instance. We see that there are two choices for each letter in this sequence (H or T), so there are a total of 210 possible such sequences. Thus, the probability of tossing 7 heads is 10C7/210 = 120 / 1024 = 15 / 128 ≈ 0.117, or 11.7%.

It is instructive to plot a histogram of the probabilities of getting a certain number of tosses of heads out of 10 tosses. This is just a bar graph, with probability on the y-axis and number of heads on the x-axis. Such a histogram can easily be constructed by dividing the entries in the 11th row of Figure 10.5.3 (Pascal's Triangle), by 210 = 1024. First we tabulate the data, then we make the histogram based on this data. The variable k stands for the number of heads and P(k) for the probability of tossing k heads out of 10 tosses. All probabilities shown are rounded to the nearest thousandth.

 

k P(k)
0
0.001
1
0.010
2
0.044
3
0.117
4
0.205
5
0.246
6
0.205
7
0.117
8
0.044
9
0.010
10
0.001
total
1.000

 

Note that the probabilities add up to 1. This is always the case, since something must happen, and we have listed all possible outcomes.

histogram of probability distribution of heads for 10 tosses of a fair coin

Figure 10.7.1: Histogram of Probability Distribution of Heads for 10 Tosses of a Fair Coin

 

As you can see, it is most likely that the number of heads will be near 5, which is half the total number of tosses. This histogram has a roughly bell-shape, which is typical of histograms of probability distributions of repeated independent trials.

Now let us see what happens as we increase the number of coin tosses. Suppose we toss a fair coin 20 times. We can follow the same procedure to determine the probability of tossing k heads. Sparing the reader the details of the calculation, we simply present the resulting histogram.

histogram of probability distribution of heads for 20 tosses of a fair coin

Figure 10.7.2: Histogram of Probability Distribution of Heads for 20 Tosses of a Fair Coin

 

Note that this histogram is narrower than the preceding one. Thus, as the number of tosses grows, the probability of the percentage of heads being close to 50% also grows. Also, note that this histogram has an even more pronounced bell shape. There is a famous result of probability theory known as the central limit theorem which says that as the number of independent trials grow, the probability distribution of any repeated experiment, such as tossing a fair coin, approaches a curve known as a normal or Gaussian distribution. Gaussian distributions occur frequently in many areas of science, including psychology, sociology, and economics.

 

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