Chapter 5.3 Independence and the Multiplication Rule

In airline​ applications, failure of a component can result in catastrophe. As a​ result, many airline components utilize something called triple modular redundancy. This means that a critical component has two backup components that may be utilized should the initial component fail. Suppose a certain critical airline component has a probability of failure of 0.0056 and the system that utilizes the component is part of a triple modular redundancy. ​(a) Assuming each​ component's failure/success is independent of the​ others, what is the probability all three components​ fail, resulting in disaster for the​ flight? ​(b) What is the probability at least one of the components does not​ fail?

(a) 0.0000018 (b) 0.9999982

About 8​% of the population of a large country is hopelessly romantic. (a) If two people are randomly​ selected, what is the probability both are hopelessly romantic​? (b) What is the probability at least one is hopelessly romantic​?

(a) 0.0064 (b) 0.1536

(a) What is the probability of obtaining two tails in a row when flipping a​ coin? Interpret this probability. (b) Interpret this probability. Consider the event of a coin being flipped two times. If that event is repeated ten thousand different​ times, it is expected that the event would result in two tails about ____ ​time(s).

(a) 0.25000 (b) 2500

The probability that a randomly selected 3​-year-old male salamander will live to be 4 years old is 0.98288. ​(a) What is the probability that two randomly selected 3​-year-old male salamanders will live to be 4 years​ old? ​(b) What is the probability that seven randomly selected 3​-year-old male salamanders will live to be 4 years​ old?

(a) 0.96605 (b) 0.88614

For a parallel structure of identical​ components, the system can succeed if at least one of the components succeeds. Assume that components fail independently of each other and that each component has a 0.18 probability of failure. (a) Would it be unusual to observe one component​ fail? Two​ components? (b) What is the probability that a parallel structure with 2 identical components will​ succeed? (c) How many components would be needed in the structure so that the probability the system will succeed is greater than 0.9998​?

(a) It WOULD NOT be unusual to observe one component​ fail, since the probability that one component​ fails, 0.18​, is GREATER than 0.05. It WOULD be unusual to observe two components​ fail, since the probability that two components​ fail, 0.0324​, is LESS than 0.05 (b) 0.9676 (c) 5

According to a​ poll, about 12​% of adults in a country bet on professional sports. Data indicates that 46.6​% of the adult population in this country is male. (a) Are the events​ "male" and​ "bet on professional​ sports" mutually​ exclusive? Explain. ​(b) Assuming that betting is independent of​ gender, compute the probability that an adult from this country selected at random is a male and bets on professional sports. (c) Using the result in part​ (b), compute the probability that an adult from this country selected at random is male or bets on professional sports. (d) The poll data indicated that 7.2​% of adults in this country are males and bet on professional sports. What does this indicate about the assumption in part​ (b)? ​(e) How will the information in part​ (d) affect the probability you computed in part​ (c)? Select the correct choice below and fill in any answer boxes within your choice.

(a) No. A person can be both male and bet on professional sports at the same time. (b) P(male and bets on professional ​sports)= 0.0559 (c) P(male or bets on professional ​sports)= 0.5301 (d) The assumption was incorrect and the events are not independent. (e) P(males or bets on professional ​sports)= 0.5140

Suppose George loses 29​% of all ping-pong games. ​(a) What is the probability that George loses two ping-pong games in a​ row? ​(b) What is the probability that George loses six ping-pong games in a​ row? ​(c) When events are​ independent, their complements are independent as well. Use this result to determine the probability that George loses six ping-pong games in a​ row, but does not lose seven in a row.

(a) P(loses 2 games)= 0.0841 (b) P(loses 6 games)= 0.0006 (c) P(loses 6 then wins 7th)= 0.0004

Determine if the following statement is true or false. When two events are​ disjoint, they are also independent.

False

Multiplication Rule for Independent Events

If E and F are independent events, then P(E and F)=P(E)⋅P(F)

Multiplication Rule for n Independent Events

If E1,E2,E3,...,En are independent events, then P(E1 and E2 and E3 and ... and En)=P(E1)⋅P(E2)⋅...⋅P(En)

The probability that a randomly selected female aged 60 years will survive the year is 0.99186 according to the National Vital Statistics Report. What is the probability that at least one of 500 randomly selected 60-year-old females will die during the course of the year?

P(at least one dies) =1−P(none die) =1−P(1st survives and 2nd survives and ⋯ and 500th survives) =1−P(1st survives)⋅P(2nd survives)⋅⋯⋅P(500th survives) =1−(0.99186)500≈1−0.0168=0.9832 If we randomly selected 500 females 60 years of age 1000 different times, we would expect at least one to die in 983 of the samples.

Suppose that events E and F are​ independent, P(E)=0.3​, and P(F)=0.6. What is the P(E and F)​?

The probability P(E and F) is 0.18

independent Event

Two events E and F are independent is the occurrence of event E in a probability experiment does not affect the probability of event F.

Dependent Event

Two events are dependent if the occurrence of event E in a probability experiment affects the probability of event F

A test to determine whether a certain antibody is present is 99.7​% effective. This means that the test will accurately come back negative if the antibody is not present​ (in the test​ subject) 99.7​% of the time. The probability of a test coming back positive when the antibody is not present​ (a false​ positive) is 0.003. Suppose the test is given to six randomly selected people who do not have the antibody. ​(a) What is the probability that the test comes back negative for all six ​people? ​(b) What is the probability that the test comes back positive for at least one of the six ​people?

​(a) P(all 6 tests are negative)= 0.9821 ​(b) P(at least one positive)= 0.0179

Disjoint Events versus Independent Events

Disjoint events and independent events are different concepts. -Recall that two events are disjoint if they have no outcomes in common, that is, if knowing that one of the events occurs, we know that the other event did not occur. -Independence means that one event occurring does not affect the probability of the other event occurring. Therefore, knowing that two events are disjoint means that the events are not independent.