question archive 1) A man and a woman get married today

1) A man and a woman get married today

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1) A man and a woman get married today. The probabilities that the husband and the wife will be alive in 40 years from today are given by 0.8 and 0.9, respectively. Assume the events that the husband and the wife will be alive in 40 years from now are independent. 

(a) Determine the probability that in 40 years both will be alive, simplify your answer.

 

(b) Determine the probability that in 40 years at least one will be alive, simplify your answer. 

 

2) a) There are 8 identical bags. In each bag, there are 8 balls, numbered from 1 to 8, inclusive. We randomly pick a ball from each bag. Determine the probability that the product of the 8 numbers on the 8 balls is a multiple of 3. You do not need to simplify your answer.

 

b) Assuming 10? bits are independently transmitted over a network in which the probability of an erroneous bit is 10??, calculate the probability of bit error when the total number of errors is less than or equal to 4. You do not need to simplify your answer. 

 

3) A box contains 26 cookies of which ???? are bad. Two cookies are eaten one by one. Assuming the probability that both cookies are good is 0.28, determine the value of ????. You must show your work and simplify your answer, that is solve the equation to obtain the integer ????. 

 

4) In assessing the strength of evidence in a legal investigation, a police detective always approaches his two informers to get information. The detective gets his information 80% of the time from the informer ???? who tells a lie 75% of the time and 20% of the time from the informer ???? who tells a lie 60% of the time. Suppose the information the detective has received is truthful, determine the probability that the information was received from the informer ????? Show your work. 

 

5) Consider a six-sided cube-shaped die, which is not fair. Let ???? be the random variable which represents the outcome of a roll of the die. The pmf of the random variable ???? is as follows: P(X=1)=P(X=2)=P(X=4)=P(X=5)=P(X=6)0.1 & P(X=3)=0.5. Determine the variance of the random variable ????. Show your work. Do simplify your answer to 2 decimal places.

 

6) a) Determine the total number of 4-letter words using the letters from A to Z inclusive, while noting that the words do not need to have meaning, and a letter cannot be used more than once in a word. You do not need to simplify your answer.

 

b) In the 6/62 lottery, the player picks 6 different numbers from a possible set of 62 different numbers, ranging from 1 to 62, inclusive. At a lottery drawing, 6 different balls are drawn at random (i.e., with equal probability) from a device containing the 62 balls representing the 62 different numbers. Note that repetition is allowed, i.e., once a ball is drawn, it is put back in the device. Determine the total number of possible combinations in this lottery. You do not need to simplify your answer.

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