Chapter 5, "Probability"

If two events are mutually exclusive, P(A and B) =

0

In a standard deck of cards, the probability of drawing an Ace is 4/52. If you draw an Ace the first time and don't replace it, the probability of drawing an Ace the second time is 3/52.

False

The two kinds of probability:

(1) Classical probability is when each outcome is equally likely to happen. What is the probability of rolling a 5 using a standard number cube? With one 5 and 6 possible outcomes, the probability is 1 over 6. (2) Empirical probability is based on observations of probability experiments. If a number cube is rolled 100 times, and a 5 is rolled 15 times, the empirical probability of rolling a 5 is 15 over 100 space equals space 3 over 20.

Law Of Large Numbers

A law that states that if an experiment is performed repeatedly, the empirical probability of an event will be close to its theoretical or actual probability. -According to the law of large numbers, if an experiment is performed repeatedly, the empirical probability of an event will be close to its theoretical or actual probability.

Probability Experiments

A trial through which specific results or outcomes are obtained.

Understanding Mutually Exclusive Events

Events that are mutually exclusive cannot occur together, that is, they have no outcomes in common. Events that can occur together are not mutually exclusive events. Consider a situation in which a university has both psychology and business departments. Some students are psychology majors only, others are business majors only, but there are also students who are both psychology and business majors. Thus, psychology and business majors are not mutually exclusive; a college student could be both at the same time.

Cards and Coins

Now consider a normal deck of 52 playing cards. You can either draw a red card or a black card; a card cannot be both red and black. Therefore, red and black playing cards are mutually exclusive; you will never have a card that fits in both categories. Another example takes us back to our coin, with heads on one side and tails on the other. When we flip a coin, only one side will face up. Thus, heads and tails are mutually exclusive; they cannot appear at the same time when flipping only one coin.

Calculate the probability that the next flip of the coin will be both heads and tails. Which statement is correct?

The probability is 0 because these events are mutually exclusive.

Complement Of An Event

The set of all outcomes in a sample space that is not included in the event E;. It is denoted as E', pronounced E prime. -It is denoted as E' (pronounced E prime), and its probability is calculated as follows:P(E) = 1 - (P(E). For example, if there is a 60% probability of rain tonight, then the probability of no rain will be: P(no rain) = 1 - 0.6 = 0.4.

What is the probability that a card drawn from a regular deck of playing cards will be a club or a spade?

0.5

An event consists of one or more outcomes, and is a subset of the total number of possible events that could occur.

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Conditional probability is the probability of an event B occurring, given that another event A has already occurred.

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One event does not affect the probability of another event occurring Independent Event --------------------- One event affects the probability of another event occurring Dependent Event ------------------- The probability of an event B occurring, given that another event A has already occurred Conditional Probability ------------------------- Used to determine the probability of occurrence of two events A and B in sequence Multiplication Rule ---------------------

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The set of all possible outcomes in a probability experiment is referred to as the sample space.

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Use the multiplication rule to determine the probability of events A and B occurring in sequence. If A and B are independent events:P(A and B) = P(A) · P(B) If A and B are dependent events:P(A and B) = P(A) · P(B/A)

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When considering the possibility of two events occurring, the events are independent if one event does not affect the probability of the other event. However, the events are dependent if one event does affect the probability of the other event

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The Addition Rule

A rule used to determine the probability of the occurrence of event A or B. -The addition rule is used to find the probability of the occurrence of event A or B. Mathematically, the addition rule is represented as follows: P(A or B) = P(A) + P(B) - P(A and B) If events are mutually exclusive, then the P(A and B) = 0. For example, what is the probability that the next card we draw from a full deck of cards will be a heart or a diamond?

Multiplication Rule

A rule used to determine the probability of the occurrence of two events A and B in sequence. -The multiplication rule is used to determine the probability of occurrence of two events A and B in sequence. In other words, we use it to answer the question, "What is the probability of A and B occurring together?" The formula for the multiplication rule is represented as follows: A and B must be independent events that can occur together. If A and B are dependent events, the multiplication rule is: P(A) * P(B/A) = P(A and B)

Range of probabilities Rule

According to this rule, the probability of an event E is always between 0 and 1. Probabilities close to 0 indicate that the event is not likely to happen; probabilities close to 1 indicate that the event is very likely to happen. A probability of .5 indicates that an event is as likely to happen as it is unlikely to happen.

Mutually exclusive

Events are mutually exclusive if they cannot occur together. Think of a mutually exclusive relationship. -Mutually exclusive events have no outcomes in common. For example, an on/off switch has two possible outcomes: on or off. It cannot be both on and off. Use the addition rule to find the probability of the occurrence of event A or B. P(A or B) = P(A) + P(B) - P(A and B)

Dependent Events

Events that are not independent are considered dependent, that is, the probability of one event occurring is dependent upon another event. Conditional probability is the probability of an event B occurring, given that another event A has already occurred. It is denoted by P(B/A). For example, if you were to consider the odds of drawing a King from a deck of cards, the probability would be 4 over 52 . If you drew a King the first time and did not replace the cad, the probability of drawing a King again would now be 3 over 51. There is one less card in the deck and one less King as well. However, if you did not draw a King the first time, the probability of drawing a King the second time (again without replacing the first card you drew) would be 4 over 51 , your odds would continue to increase until you drew a King.

Probability

Probability is the likelihood that an event will happen. An event is made up of one or more outcomes and is part of the total number of possible events that could occur. - You can calculate probability if you know the number of outcomes of an event and the total number of possible outcomes.

Ahmed's grandfather has a jar of marbles and a jar of coins, and he allows Ahmed to take one item from each jar. The jar of marbles contains 30 marbles in total; 20 are red, 7 are blue, and 3 are yellow. The jar of coins contains 10 coins in total; 6 are pennies, 3 are dimes, and 1 is a quarter. If Ahmed chooses a coin and a marble at random, the probability that he will choose a yellow marble and a quarter is .01.

True

Independent Events

Two or more events such that the occurrence of one of the events does not affect the probability of occurrence of the others.

Understanding Independent and Dependent Events

When one event does not affect the probability of occurrence of another event, the two events are considered independent. For example, getting a 2 after rolling a die and drawing an Ace from a deck of cards are independent events. In our previous probability examples, we were only dealing with one event. We only wanted to know the probability of some event, E, occurring. Often, we will have two events that we need to consider when calculating probability. Calculating the probability of A and B occuring together requires that we know what types of events are being considered: independent or dependent events.

Empirical or statistical probability

is based on observations obtained from probability experiments.

Classical or theoretical probability

refers to the type of probability when each outcome is equally likely to occur.