Section 5.1 Basic Ideas

Objectives Construct sample spaces Compute and interpret probabilities Approximate probabilities using the Empirical Method Approximate probabilities by using Simulation

Objective 1 Construct sample spaces

Probability Experiment A probability experiment is one in which we do not know what any individual outcome will be, but we do know how a long series of repetitions will come out. For example, if we toss a fair coin, we do not know what the outcome of a single toss will be, but we do know what the outcome of a long series of tosses will be – about half “heads” and half “tails”.

Probability The probability of an event is the proportion of times that the event occurs in the long run. So, for a “fair” coin, that is, one that is equally likely to come up heads as tails, the probability of heads is 1/2 and the probability of tails is 1/2.

Law of Large Numbers The law of large numbers says that as a probability experiment is repeated again and again, the proportion of times that a given event occurs will approach its probability.

Sample Space The collection of all the possible outcomes of a probability experiment is called a sample space. Example: Suppose that a coin is tossed. The sample space consists of: {Heads, Tails} Suppose that a standard die is rolled. The sample space consists of: {1, 2, 3, 4, 5, 6}

Event We are often concerned with occurrences that consist of several outcomes. For example, when rolling a die, we might be concerned with the possibility of rolling an odd number. A collection of outcomes of a sample space is called an event. Example: A probability experiment consists of rolling a die. The sample space is {1, 2, 3, 4, 5, 6}. The event of rolling an odd number = {1, 3, 5}.

Probability Model Once we have a sample space for an experiment, we need to specify the probability of each event. This is done with a probability model. We use the letter “P” to denote probabilities. For example, if we toss a coin, we denote the probability that the coin lands heads by “P(Heads).” Notation: If A denotes an event, the probability of event A is denoted by P(A).

Probabilities With Equally Likely Outcomes If a sample space has n equally likely outcomes, and an event A has k outcomes, then

Objective 2 Compute and interpret probabilities

Example A fair die is rolled. Find the probability that an odd number comes up. Solution: The sample space has six equally likely outcomes: {1, 2, 3, 4, 5, 6} The event of an odd number has three outcomes: {1, 3, 5} The probability is:

Example A family has three children. Denoting a boy by B and a girl by G, we can denote the genders of these children from oldest to youngest. For example, GBG means the oldest child is a girl, the middle child is a boy, and the youngest child is a girl. There are eight possible outcomes: BBB, BBG, BGB, BGG, GBB, GBG, GGB, and GGG. Assume these outcomes are equally likely. What is the probability that all three children are the same gender? Solution: Of the eight equally likely outcomes, the two outcomes BBB and GGG correspond to having all children of the same gender. Therefore

Probability Rules The probability of an event is always between 0 and 1. That is, 0 ≤ P(A) ≤ 1. If A cannot occur, then P(A) = 0. If A is certain to occur, then P(A) = 1.

Sampling From a Population Sampling an individual from a population is a probability experiment. The population is the sample space and members of the population are equally likely outcomes. Example: There are 10,000 families in a certain town categorized as follows: A pollster samples a single family from this population. What is the probability that the sampled family rents? Solution: The number of families who rent is 912 + 2857 = 3769. Therefore, the probability that the sampled family rents is 3769/10,000 = 0.3769. Own a house Own a condo Rent a house Rent an apartment 4753 1478 912 2857

Unusual Events An unusual event is one that is not likely to happen. In other words, an event whose probability is small. A rule of thumb is that any event whose probability is less than 0.05 is considered to be unusual. Example: In a college of 5000 students, 150 are math majors. A student is selected at random and turns out to be a math major. Is this an unusual event? Solution: The event of choosing a math major consists of 150 students out of a total of 5000 students. The probability of choosing a math major is 150/5000 = 0.03. Since 0.03 < 0.05, this would be considered an unusual event.

Objective 3 Approximate probabilities using the Empirical Method

Approximating Probabilities with the Empirical Method The law of large numbers says that if we repeat a probability experiment a large number of times, then the proportion of times that a particular outcome occurs is likely to be close to the true probability of the outcome. The Empirical Method consists of repeating an experiment a large number of times, and using the proportion of times an outcome occurs to approximate the probability of the outcome.

Example The Centers for Disease Control reports that in the year 2002 there were 2,057,979 boys and 1,963,747 girls born in the U.S. Approximate the probability that a newborn baby is a boy. Solution: The number of times that the experiment has been repeated is: 2,057,979 boys + 1,963,747 girls = 4,021,726 births The proportion of births that are boys is: 2,057,979/4,021,726 = 0.5117 Therefore, the probability that a newborn baby is a boy is approximated by 0.5117.

Objective 4 Approximate probabilities by using Simulation

Simulation In practice, it can be difficult or impossible to repeat an experiment many times in order to approximate a probability with the Empirical Method. In some cases, we can use technology to repeat an equivalent virtual experiment many times. Conducting a virtual experiment in this way is called simulation.

Example If three dice are rolled, the smallest possible total is 3, and the largest possible total is 18. We will perform a simulation to estimate the probability that the sum of three dice is 12 or less. If we had no technology, we could roll three dice a large number of times and compute the proportion of times that the sum was 12 or less. With technology, we can use a random number generator in place of dice. For this example, we will show how to use the software package MINITAB to simulate rolling three dice 1000 times.

Solution Step 1: Click Calc→Random Data→Integer. Since we want 1000 replications of the rolling of three dice, we specify 1000 rows of data, columns C1–C3, a minimum value of 1, and a maximum value of 6. Click OK. Step 2: Click Calc→Row Statistics. We fill out the dialog boxes with C1, C2, C3 and click OK. Note: C1, C2, and C3 are the columns that contains the results of rolling three dice 1000 times. The sum will be placed in column, C4.

Solution Step 3: Click Stat→Tables→Tally. We specify column C4 and cumulative counts as shown in the figure. Click OK. This produces the table shown below. The table shows that 745 of the 1000 rolls of the dice produced a number of 12 or less. We estimate the probability of obtaining a total of 12 or less in a roll of three dice to be P(12 or less) = 745/1000 ≈ 0.745

Do You Know… How to construct a sample space? How to compute probabilities of equally likely events? The rules of probability? How to compute probabilities using the Empirical Method? How to approximate probabilities by using simulation?