State whether the following statements are True or False. Give a reason in support of your answer.
(a) If events A, B and C are three mutually exclusive and exhaustive events, then it is possible that ,
and
.
(b) If X is a discrete random variable, then X cannot take countably infinite values.
(c) If X is a standard normal variate, then .
(d) If random variables X and Y follow Bernoulli distributions with parameters 1/2 and 1/3 respectively then the random variable X + Y also follows a Bernoulli distribution with parameter 5/6.
(a) True – If A, B, and C are mutually exclusive and exhaustive, then (P(A) + P(B) + P(C) = 1). Equations involving probabilities like (P(A \cup B) = _______ ______ ___ __________ __________.
______ __________ ________ _________ __________ _______ ______ _________ _______ __________.
___ _____ _________ ___ ____ ________ _________ ____.
_______ ________ _________ __________ ____ ______ _______ ___ _______.
_____ ___ ______ ______ _________ _________.
_________ _________ _____ ___ _________ _______ ______.
_____ ___ _____ __________ _________ ________ ___ ____ ____ _______.
______ ____ _________ _____ ____.
____ ___ __________ __________ ____ ___ ________ ______ ______ __________ _____ _________.
__________ ______ ________ _______ __________ ____ _________.
______ ______ _____ _______ ______ ____ _____ _______ __________ __________.
_______ ________ ______ _____ __________.
_____ ______ __________ ____ __________ _____.
_____ ______ _____ __________ __________ ________ _____.
___ _________ ____ ______ _____ ________ _______ _________ ___ ____.
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Three unbiased coins are tossed simultaneously. In which of the following cases are the events A and B independent?
(i) A be the event of getting exactly one tail
B be the event of getting exactly one head
(ii) A be the event that first coin shows tail
B be the event that third coin shows head
If a problem is randomly selected from a particular book, then probability that students A, B and C can solve the problem are 1/2, 1/3 and 1/4 respectively. If a problem is selected randomly from this book and given to the students A, B and C then what is the probability that problem is solved. They solve the problem independently.
Find the probability that a third child in a family is the family's second daughter, assuming the male and female are equally probable.
Four coins were tossed and number of heads noted. The experiment is repeated 200 times. The number of tosses showing 0, 1, 2, 3 and 4 heads were found distributed as under. Fit a binomial distribution to these observed results assuming that the nature of the coins is not known.
| Number of Heads | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| Number of Tosses | 15 | 35 | 90 | 40 | 20 |
Two cards are drawn successively with replacement from a well shuffled pack of 52 cards. Find the expected value for the number of kings.
State whether the following statements are True or False. Give a reason in support of your answer.
(a) If events A, B and C are three mutually exclusive and exhaustive events, then it is possible that ,
and
.
(b) If X is a discrete random variable, then X cannot take countably infinite values.
(c) If X is a standard normal variate, then .
(d) If random variables X and Y follow Bernoulli distributions with parameters 1/2 and 1/3 respectively then the random variable X + Y also follows a Bernoulli distribution with parameter 5/6.
A factory produces certain type of output by 3 machines. The respective daily production figures are- machine X : 4500 units, machine Y: 2000 units and machine Z: 3500 units. Past experience shows that 2% of the output produced by machine X is defective. The corresponding fractions of defectives for the other two machines are 5 and 10 percent respectively. An item is drawn from the day's production. If the drawn item is found to be defective, what is the probability that it has been produced by machine Y?