Probability, Theory of Probability

When a coin is tossed, we can reason that the chance, or likelihood, that it will fall heads is 1 out of 2, or the probability that it will fall heads is 1/2. Of course, this does not mean that if a coin is tossed 10 times it will necessarily fall heads 5 times. If the coin is a “fair coin” and it is tossed a great many times, however, it will fall heads very nearly half of the time. Here we give an introduction to two kinds of probability, experimental and theoretical.

Experimental and Theoretical Probability

If we toss a coin a great number of times — say, 1000 — and count the number of times it falls heads, we can determine the probability that it will fall heads. If it falls heads 503 times, we would calculate the probability of its falling heads to be
503/1000, or 0,503.

This is an experimental determination of probability. Such a determination of probability is discovered by the observation and study of data and is quite common and very useful. Here, for example, are some probabilities that have been determined experimentally :

1. The probability that a woman will get breast cancer in her lifetime is 1/11.

2. If you kiss someone who has a cold, the probability of your catching a cold is 0,07.

3. A person who has just been released from prison has an 80% probability of returning.

If we consider a coin and reason that it is just as likely to fall heads as tails, we would calculate the probability that it will fall heads to be 1/2. This is a theoretical determination of probability. Here are some other probabilities that have been determined theoretically, using mathematics:

1. If there are 30 people in a room, the probability that two of them have the same birthday (excluding year) is 0,706.

2. While on a trip, you meet someone and, after a period of conversation, discover that you have a common acquaintance. The typical reaction, “It’s a small world!”, is actually not appropriate, because the probability of such an occurrence is quite high—just over 22%.

In summary, experimental probabilities are determined by making observations and gathering data. Theoretical probabilities are determined by reasoning mathematically. Examples of experimental and theoretical probability like those above, especially those we do not expect, lead us to see the value of a study of probability. You might ask, “What is the true probability?” In fact, there is none. Experimentally, we can determine probabilities within certain limits. These may or may not agree with the probabilities that we obtain theoretically. There are situations in which it is much easier to determine one of these types of probabilities than the other. For example, it would be quite difficult to arrive at the probability of catching a cold using theoretical probability.

Computing Experimental Probabilities

We first consider experimental determination of probability. The basic principle we use in computing such probabilities is as follows.

Principle P (Experimental)

Given an experiment in which n observations are made, if a situation, or event, E occurs m times out of n observations, then we say that the experimental probability of the event, , is given by P(E) = m/n.

Example 1 Sociological Survey. The authors of this text conducted an experimental survey to determine the number of people who are lefthanded, right-handed, or both. The results are shown in the graph at left.

a) Determine the probability that a person is right-handed.

b) Determine the probability that a person is left-handed.

c) Determine the probability that a person is ambidextrous (uses both hands with equal ability).

d) There are 120 bowlers in most tournaments held by the Professional Bowlers Association. On the basis of the data in this experiment, how many of the bowlers would you expect to be left-handed?

Solution

a) The number of people who are right-handed is 82, the number who are left-handed is 17, and the number who are ambidextrous is 1. The total number of observations is , or 100. Thus the probability that a person is right-handed is P, where
P = 82/100, or 0,82, or 82%.

b) The probability that a person is left-handed is P, where
P = 17/100, or 0,17, or 17%.

c) The probability that a person is ambidextrous is P, where
P = 1/100, or 0,01, or 1%.

d) There are 120 bowlers, and from part (b) we can expect 17% to be left-handed. Since
17% of 120 = 0,17.120 = 20,4,
we can expect that about 20 of the bowlers will be left-handed.

Example 2 Quality Control. It is very important for a manufacturer to maintain the quality of its products. In fact, companies hire quality control inspectors to ensure this process. The goal is to produce as few defective products as possible. But since a company is producing thousands of products every day, it cannot afford to check every product to see if it is defective. To find out what percentage of its products are defective, the company checks a smaller sample.
The U.S. Department of Agriculture requires that 80% of the seeds that a company produces sprout. To determine the quality of the seeds it produces, a company takes 500 seeds from those it has produced and plants them. It finds that 417 of the seeds sprout.

a) What is the probability that a seed will sprout?

b) Did the seeds meet government standards?

Solutiona) We know that 500 seeds were planted and 417 sprouted. The probability of a seed sprouting is P, where
P = 417/500 = 0,834, or 83.4%.

b) Since the percentage of seeds that sprouted exceeded the 80% requirement, the seeds meet government standards.

Example 3 Television Ratings. There are an estimated 105,500,000 households with televisions in the United States. Each week, viewing information is collected and reported. In one week, 7,815,000 households tuned in to the popular comedy series “Everybody Loves Raymond” on CBS and 8,302,000 households tuned in to the popular drama series “Law and Order” on NBC (Source: Nielsen Media Research). What is the probability that a television household tuned in to “Everybody Loves Raymond” during the given week? to “Law and Order”?

Solution The probability that a television household was tuned in to “Everybody Loves Raymond” is P, where
P = 7,815,000/105,500,000 ≈ 0,074 ≈ 7,4%.
The probability that a television household was tuned in to “Law and Order” is P, where
P = 8,302,000/105,500,000 ≈ 0,079 ≈ 7,9%.
These percentages are called ratings.

Theoretical Probability

Suppose that we perform an experiment such as flipping a coin, throwing a dart, drawing a card from a deck, or checking an item off an assembly line for quality. Each possible result of such an experiment is called an outcome. The set of all possible outcomes is called the sample space. An event is a set of outcomes, that is, a subset of the sample space.

Example 4 Dart Throwing. Consider this dartboard. Assume that the experiment is “throwing a dart” and that the dart hits the board. Find each of the following.

a) The outcomes

b) The sample space

Solution
a) The outcomes are hitting black (B), hitting red (R), and hitting white (W).

b) The sample space is {hitting black hitting red, hitting white}, which can be simply stated as {B, R, W}.

Example 5 Die Rolling. A die (pl., dice) is a cube, with six faces, each containing a number of dots from 1 to 6 on each side.

Suppose a die is rolled. Find each of the following.
a) The outcomes
b) The sample space

Solution
a) The outcomes are 1, 2, 3, 4, 5, 6.
b) The sample space is {1, 2, 3, 4, 5, 6}.

We denote the probability that an event E occurs as P(E). For example, “a coin falling heads” may be denoted H. Then P(H) represents the probability of the coin falling heads. When all the outcomes of an experiment have the same probability of occurring, we say that they are equally likely. To see the distinction between events that are equally likely and those that are not, consider the dartboards shown below.

For board A, the events hitting black, hitting red, and hitting white are equally likely, because the black, red, and white areas are the same. However, for board B the areas are not the same so these events are not equally likely.

Principle P (Theoretical)

If an event E can occur m ways out of n possible equally likely outcomes of a sample space S, then the theoretical probability of the event, P(E), is given by
P(E) = m/n.

Example 6 What is the probability of rolling a 3 on a die?

Solution On a fair die, there are 6 equally likely outcomes and there is 1 way to roll a 3. By Principle P, P(3) = 1/6.

Example 7 What is the probability of rolling an even number on a die?

Solution The event is rolling an even number. It can occur 3 ways (getting 2, 4, or 6). The number of equally likely outcomes is 6. By Principle P, P(even) = 3/6,or 1/2.

We will use a number of examples related to a standard bridge deck of 52 cards. Such a deck is made up as shown in the following figure.

Example 8 What is the probability of drawing an ace from a wellshuffled deck of cards?

Solution There are 52 outcomes (the number of cards in the deck), they are equally likely (from a well-shuffled deck), and there are 4 ways to obtain an ace, so by Principle P, we have
P(drqawing an ace) = 4/52, or 1/13.

Example 9 Suppose that we select, without looking, one marble from a bag containing 3 red marbles and 4 green marbles. What is the probability of selecting a red marble?

Solution There are 7 equally likely ways of selecting any marble, and since the number of ways of getting a red marble is 3, we have
P(selecting a red marble) = 3/7.

The following are some results that follow from Principle P.

Probability Properties

a) If an event E cannot occur, then P(E) = 0.
b) If an event E is certain to occur, then P(E) = 1.
c) The probability that an event E will occur is a number from 0 to 1: 0 ≤ P(E) ≤ 1.

For example, in coin tossing, the event that a coin will land on its edge has probability 0. The event that a coin falls either heads or tails has probability 1.

Example 10 Suppose that 2 cards are drawn from a well-shuffled deck of 52 cards. What is the probability that both of them are spades?

Solution The number of ways n of drawing 2 cards from a well-shuffled deck of 52 cards is 52C2. Since 13 of the 52 cards are spades, the number of ways m of drawing 2 spades is 13C2. Thus,
P(getting 2 spades) = m/n = 13C2/52C2 = 78/1326 = 1/17.

Example 11 Suppose that 3 people are selected at random from a group that consists of 6 men and 4 women. What is the probability that 1 man and 2 women are selected?

Solution The number of ways of selecting 3 people from a group of 10 is 10C3. One man can be selected in 6C1 ways, and 2 women can be selected in 4C2 ways. By the fundamental counting principle, the number of ways of selecting 1 man and 2 women is 6C1.4C2. Thus the probability that 1 man and 2 women are selected is
P = 6C1.4C2/10C3 = 3/10.

Example 12 Rolling Two Dice. What is the probability of getting a total of 8 on a roll of a pair of dice?

Solution On each die, there are 6 possible outcomes. The outcomes are paired so there are 6.6, or 36, possible ways in which the two can fall. (Assuming that the dice are different—say, one red and one blue—can help in visualizing this.)

The pairs that total 8 are as shown in the figure above. There are 5 possible ways of getting a total of 8, so the probability is 5/36.


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