Elementary Statistics W3L7

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 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).

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.

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}

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?

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.

The 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.

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.

There are 10,000 families in a certain town categorized as follows: Own a house-4753, Own a condo-1478, Rent a house-912, Rent an apartment-2857 A pollster samples a single family from this population. What is the probability that the sampled family rents?

The number of families who rent is 912 + 2857 = 3769. Therefore, the probability that the sampled family rents is 3769/10,000 = 0.3769

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.

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.

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.

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}

Unusual event

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.

A fair die is rolled. Find the probability that an odd number comes up.

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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?

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Probability Rules

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