Introduction to Bernoulli trial probability
Bernoulli Trails probability: Trails of a random experiment are called Bernoulli trails, if they satisfy the following conditions:
(i) There should be a finite number of trails.
(ii) The trails should be independent.
(iii) Each trail has exactly two outcomes: success or failure.
(iv) The probability of success remains the same in each trail.
Examples on Bernoulli Trial Probability
For example : Throwing a die 50 times is a case of 50 Bernoulli trails, in which each trail results in the success (say an even number ) or the failure ( an odd number ) and the probability of success (p) is same for all 50 throws. Obviously, the successive throws of the die is independent experiments. If the die is fair and have six numbers 1 to 6 written on six faces, then p =`(1)/(2)` and q=1-p = 1-`(1)/(2)` =`(1)/(2)` = probability of failure.
Solved Problems on Bernoulli Trial Probability
Question based on Bernoulli trail probability:
Qu: Six balls are drawn successively from an urn containing 7 red and 9 black balls. Tell whether or not the trails of drawing balls are Bernoulli trails when after each draw the ball is draw the ball drawn is
(i) Replaced into the urn.
(ii) Not replaced in the urn.
Solution:(i) The number of trails is finite. When the drawing is done with replacement, the probability of success (say, red ball) is p=`(7)/(16)` which is same for all six trails (draws). Hence, the drawings of balls with replacement are Bernoulli trails.
(ii) When the drawing is done without replacement, the probability of success (i.e. red ball) in first trail is `(7)/(16)` ; in 2nd trail is `(6)/(15)` if the first ball drawn is red or `(7)/(15)` if the first ball drawn is black and so on. Clearly, the probability of success is not same for all trails; hence the trails are not Bernoulli trails.
Bernoulli Trails probability: Trails of a random experiment are called Bernoulli trails, if they satisfy the following conditions:
(i) There should be a finite number of trails.
(ii) The trails should be independent.
(iii) Each trail has exactly two outcomes: success or failure.
(iv) The probability of success remains the same in each trail.
Examples on Bernoulli Trial Probability
For example : Throwing a die 50 times is a case of 50 Bernoulli trails, in which each trail results in the success (say an even number ) or the failure ( an odd number ) and the probability of success (p) is same for all 50 throws. Obviously, the successive throws of the die is independent experiments. If the die is fair and have six numbers 1 to 6 written on six faces, then p =`(1)/(2)` and q=1-p = 1-`(1)/(2)` =`(1)/(2)` = probability of failure.
Solved Problems on Bernoulli Trial Probability
Question based on Bernoulli trail probability:
Qu: Six balls are drawn successively from an urn containing 7 red and 9 black balls. Tell whether or not the trails of drawing balls are Bernoulli trails when after each draw the ball is draw the ball drawn is
(i) Replaced into the urn.
(ii) Not replaced in the urn.
Solution:(i) The number of trails is finite. When the drawing is done with replacement, the probability of success (say, red ball) is p=`(7)/(16)` which is same for all six trails (draws). Hence, the drawings of balls with replacement are Bernoulli trails.
(ii) When the drawing is done without replacement, the probability of success (i.e. red ball) in first trail is `(7)/(16)` ; in 2nd trail is `(6)/(15)` if the first ball drawn is red or `(7)/(15)` if the first ball drawn is black and so on. Clearly, the probability of success is not same for all trails; hence the trails are not Bernoulli trails.
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