8B-Probability Theory PDF

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Summary

Probability Theory...

Description

Probability Rules Probability is measured on a scale from 0 to 1: If event E is impossible then P(E) = 0 If event E is certain to occur then P(E) = 1 The sum of probabilities of all events in the sample space is 1:

∑ ( )=

Complement probability: Consists of all possible outcomes from the sample space that are not in event. Denoted by Ē where E is the event:

= −

Joint Probability: Both events A and B happen. Denoted by

.

Either event A occurs or event B occurs or they both occur. Denoted by The relationship between these two probabilities is given by:

.

=

+

−

Conditional Probability: Event A occurs given that event B has happened. Denoted by

. Conditional probabilities can be

used to determine the probability of two or more events happening in sequence:

=

×

.

If events A and B are independent then If events A and B are mutually exclusive then

=

×

.

= .

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Examples: What is the probability of selecting a card that is a Queen and a spade?

What is the probability of selecting a card that is a Queen or a spade?

What is the probability of selecting a King and then another King if the cards are not replaced after selecting them?

A bag contains 3 green marbles and 4 red marbles. What is the probability of selecting a green marble and then selecting a red marble assuming the marbles are not replaced after selecting them?

What is the probability that when a coin is flipped twice, it lands tails both times?

What is the probability of rolling a dice three times and getting at least one roll of 6 in the three rolls?

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Exercises: 1. What is the probability that when a coin is flipped six times in a row, it lands heads every time? 2. What is the probability of selecting an ace of spades or a king of spades from a standard deck of cards? 3. What is the probability of selecting an ace of spades, and then a king of spades from a standard deck of cards if the cards are not replaced after selecting them? 4. What is the probability of selecting an ace of spades, and then a king of spades, from a standard deck of cards if the cards are replaced (and shuffled) after selecting them? 5. What is the probability that one card selected from a standard deck is an ace or a heart? 6. What is the probability that a die never comes up an even number when it is rolled five times? 7. What is the probability that a randomly selected card from a standard deck is a club given that it is a black suit? 8. A box of 12 laptops contains 6 Acers, 5 Lenovos, and 1 Mac. Two laptops are selected at random from the box without replacement. a) What is the probability the first is a Mac and the second is an Acer? b) What is the probability that both are a Lenovo? c) What is the probability that an Acer and a Lenovo are selected (in any order)?

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Answers: (rounded to 4 decimal places) 1. 2. 3. 4. 5. 6. 7.

0.0156 0.0385 0.0003770...≅ 0.0004 0.0003698...≅ 0.0004 0.3077 0.03125 ≅ 0.0313 0.5000

8. a) 0.0455 b) 0.1515 c) 0.4545

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