Advertisements
Advertisements
Question
A, B, and C are independent witness of an event which is known to have occurred. Aspeaks the truth three times out of four, B four times out of five and C five times out of six. What is the probability that the occurrence will be reported truthfully by majority of three witnesses?
Solution
\[P\left(\text{ A speaks truth } \right) = \frac{3}{4}\]
\[P\left( \text{ B speaks truth } \right) = \frac{4}{5}\]
\[P\left( \text{ C speaks truth } \right) = \frac{5}{6}\]
\[P\left( \text{ majority speaks truth } \right) = P\left( \text{ two speak truth } \right) + P\left( \text{ all speak truth } \right)\]
\[ = P\left( A \right) \times P\left( B \right)\left[ 1 - P\left( C \right) \right] + P\left( A \right) \times P\left( C \right)\left[ 1 - P\left( B \right) \right] + P\left( C \right) \times P\left( B \right)\left[ 1 - P\left( A \right) \right] + P\left( A \right) \times P\left( B \right) \times P\left( C \right)\]
\[ = \frac{3}{4} \times \frac{4}{5}\left( 1 - \frac{5}{6} \right) + \frac{3}{4} \times \frac{5}{6}\left( 1 - \frac{4}{5} \right) + \frac{4}{5} \times \frac{5}{6}\left( 1 - \frac{3}{4} \right) + \frac{3}{4} \times \frac{4}{5} \times \frac{5}{6}\]
\[ = \frac{12}{120} + \frac{15}{120} + \frac{20}{120} + \frac{60}{120}\]
\[ = \frac{107}{120}\]
APPEARS IN
RELATED QUESTIONS
How many times must a fair coin be tossed so that the probability of getting at least one head is more than 80%?
A bag A contains 4 black and 6 red balls and bag B contains 7 black and 3 red balls. A die is thrown. If 1 or 2 appears on it, then bag A is chosen, otherwise bag B, If two balls are drawn at random (without replacement) from the selected bag, find the probability of one of them being red and another black.
Assume that each child born is equally likely to be a boy or a girl. If a family has two children, what is the conditional probability that both are girls given that (i) the youngest is a girl, (ii) at least one is a girl?
A bag contains 5 white, 7 red and 3 black balls. If three balls are drawn one by one without replacement, find the probability that none is red.
A card is drawn from a well-shuffled deck of 52 cards and then a second card is drawn. Find the probability that the first card is a heart and the second card is a diamond if the first card is not replaced.
If A and B are two events such that P (A ∩ B) = 0.32 and P (B) = 0.5, find P (A/B).
If A and B are two events such that \[ P\left( A \right) = \frac{7}{13}, P\left( B \right) = \frac{9}{13} \text{ and } P\left( A \cap B \right) = \frac{4}{13}, \text{ then find } P\left( \overline{ A }|B \right) . \]
Two coins are tossed once. Find P (A/B) in each of the following:
A = Tail appears on one coin, B = One coin shows head.
A card is drawn from a pack of 52 cards so the teach card is equally likely to be selected. In which of the following cases are the events A and B independent?
A = the card drawn is black, B = the card drawn is a king.
A coin is tossed three times. Let the events A, B and C be defined as follows:
A = first toss is head, B = second toss is head, and C = exactly two heads are tossed in a row.
Check the independence of A and B.
A coin is tossed three times. Let the events A, B and C be defined as follows:
A = first toss is head, B = second toss is head, and C = exactly two heads are tossed in a row. C and A
If A and B be two events such that P (A) = 1/4, P (B) = 1/3 and P (A ∪ B) = 1/2, show that A and B are independent events.
Given two independent events A and B such that P (A) = 0.3 and P (B) = `0.6. Find P (A ∩ overlineB ) `.
The odds against a certain event are 5 to 2 and the odds in favour of another event, independent to the former are 6 to 5. Find the probability that (i) at least one of the events will occur, and (ii) none of the events will occur.
A bag contains 3 red and 5 black balls and a second bag contains 6 red and 4 black balls. A ball is drawn from each bag. Find the probability that one is red and the other is black.
A husband and wife appear in an interview for two vacancies for the same post. The probability of husband's selection is 1/7 and that of wife's selection is 1/5. What is the probability that both of them will be selected ?
The probability of student A passing an examination is 2/9 and of student B passing is 5/9. Assuming the two events : 'A passes', 'B passes' as independent, find the probability of : (i) only A passing the examination (ii) only one of them passing the examination.
A, B and C in order toss a coin. The one to throw a head wins. What are their respective chances of winning assuming that the game may continue indefinitely?
Three persons A, B, C throw a die in succession till one gets a 'six' and wins the game. Find their respective probabilities of winning.
Fatima and John appear in an interview for two vacancies for the same post. The probability of Fatima's selection is \[\frac{1}{7}\] and that of John's selection is \[\frac{1}{5}\] What is the probability that
(i) both of them will be selected?
(ii) only one of them will be selected?
(iii) none of them will be selected?
The contents of three bags I, II and III are as follows:
Bag I : 1 white, 2 black and 3 red balls,
Bag II : 2 white, 1 black and 1 red ball;
Bag III : 4 white, 5 black and 3 red balls.
A bag is chosen at random and two balls are drawn. What is the probability that the balls are white and red?
When three dice are thrown, write the probability of getting 4 or 5 on each of the dice simultaneously.
If A and B are independent events such that P(A) = p, P(B) = 2p and P(Exactly one of Aand B occurs) = \[\frac{5}{9}\], then find the value of p.
If one ball is drawn at random from each of three boxes containing 3 white and 1 black, 2 white and 2 black, 1 white and 3 black balls, then the probability that 2 white and 1 black balls will be drawn is
The probabilities of a student getting I, II and III division in an examination are \[\frac{1}{10}, \frac{3}{5}\text{ and } \frac{1}{4}\]respectively. The probability that the student fails in the examination is
Out of 30 consecutive integers, 2 are chosen at random. The probability that their sum is odd, is
A bag contains 5 black balls, 4 white balls and 3 red balls. If a ball is selected randomwise, the probability that it is black or red ball is
A coin is tossed three times. If events A and B are defined as A = Two heads come, B = Last should be head. Then, A and B are ______.
Five persons entered the lift cabin on the ground floor of an 8 floor house. Suppose that each of them independently and with equal probability can leave the cabin at any floor beginning with the first, then the probability of all 5 persons leaving at different floors is
A box contains 6 nails and 10 nuts. Half of the nails and half of the nuts are rusted. If one item is chosen at random, the probability that it is rusted or is a nail is
If P (A ∪ B) = 0.8 and P (A ∩ B) = 0.3, then P \[\left( A \right)\] \[\left( A \right)\] + P \[\left( B \right)\] =
Mark the correct alternative in the following question:
\[\text{ If A and B are two independent events with } P\left( A \right) = \frac{3}{5} \text{ and } P\left( B \right) = \frac{4}{9}, \text{ then } P\left( \overline{A} \cap B \right) \text{ equals } \]
Mark the correct alternative in the following question: A bag contains 5 red and 3 blue balls. If 3 balls are drawn at random without replacement, then the probability that exactly two of the three balls were red, the first ball being red, is
Mark the correct alternative in the following question:
Two cards are drawn from a well shuffled deck of 52 playing cards with replacement. The probability that both cards are queen is
Mark the correct alternative in the following question:
\[\text{ If A and B are such that } P\left( A \cup B \right) = \frac{5}{9} \text{ and } P\left( \overline{A} \cup \overline{B} \right) = \frac{2}{3}, \text{ then } P\left( A \right) + P\left( B \right) = \]
An insurance company insured 3000 cyclists, 6000 scooter drivers, and 9000 car drivers. The probability of an accident involving a cyclist, a scooter driver, and a car driver are 0⋅3, 0⋅05 and 0⋅02 respectively. One of the insured persons meets with an accident. What is the probability that he is a cyclist?
