Advertisements
Advertisements
Question
Suppose we have four boxes. A, B, C and D containing coloured marbles as given below:
| Box | Marble colour | ||
| Red | White | Black | |
| A | 1 | 6 | 3 |
| B | 6 | 2 | 2 |
| C | 8 | 1 | 1 |
| D | 0 | 6 | 4 |
One of the boxes has been selected at random and a single marble is drawn from it. If the marble is red, what is the probability that it was drawn from box A?, box B?, box C?
Advertisements
Solution
The probability of selecting one box out of the given 4 boxes = `1/4`
i.e., P(E1) = P(E2) = P(E3) = P(E4) = `1/4`
Let the 4th event be drawing a red piece. There are a total of 10 pieces in box A, of which 1 is red.
∴ `P(A/E_1) = 1/10`
Similarly `P(A/E_2) = 6/10, P(A/E_3) = 8/10, P(A/E_4) = 0`
(i) Again, by Bayes' theorem,
`P((E_1)/A) = (P(E_1) xx P((A)/E_1))/(P(E_1) xxP(A/E_1) + P(E_2) xx P(A/E_2) + P(E_3) xx P(A/E_3) + P(E_4) xx P(A/E_4))`
= `(1/4 xx 1/10)/(1/4 xx 1/10 + 1/4 xx 6/10 + 1/4 xx 8/10 + 1/4 xx 0)`
= `1/(1 + 6 + 8)`
= `1/15`
(ii) Again, by Bayes' theorem,
= `P((E_2)/A) = (P(E_2) xx P((A)/E_2))/(P(E_1) xxP(A/E_1) + P(E_2) xx P(A/E_2) + P(E_3) xx P(A/E_3) + P(E_4) xx P(A/E_4))`
= `(1/4 xx 6/10)/(1/4 xx 1/10 + 1/4 xx 6/10 + 1/4 xx 8/10 + 1/4 xx 0)`
= `6/(1 + 6 + 8)`
= `6/15`
= `2/5`
(iii) and by Bayes' theorem,
= `P((E_3)/A) = (P(E_3) xx P((A)/E_3))/(P(E_1) xxP(A/E_1) + P(E_2) xx P(A/E_2) + P(E_3) xx P(A/E_3) + P(E_4) xx P(A/E_4))`
= `(1/4 xx 8/10)/(1/4 xx 1/10 + 1/4 xx 6/10 + 1/4 xx 8/10 + 1/4 xx 0)`
= `8/(1 + 6+ 8)`
= `8/15`
Hence, the probability of a red piece being selected from box A, box B, and box C is `1/5`, `2/5` and `8/15`, respectively.
APPEARS IN
RELATED QUESTIONS
Assume that the chances of a patient having a heart attack is 40%. Assuming that a meditation and yoga course reduces the risk of heart attack by 30% and prescription of certain drug reduces its chance by 25%. At a time a patient can choose any one of the two options with equal probabilities. It is given that after going through one of the two options, the patient selected at random suffers a heart attack. Find the probability that the patient followed a course of meditation and yoga. Interpret the result and state which of the above stated methods is more beneficial for the patient.
In a game, a man wins Rs 5 for getting a number greater than 4 and loses Rs 1 otherwise, when a fair die is thrown. The man decided to thrown a die thrice but to quit as and when he gets a number greater than 4. Find the expected value of the amount he wins/loses
40% students of a college reside in hostel and the remaining reside outside. At the end of the year, 50% of the hostelers got A grade while from outside students, only 30% got A grade in the examination. At the end of the year, a student of the college was chosen at random and was found to have gotten A grade. What is the probability that the selected student was a hosteler ?
A bag X contains 4 white balls and 2 black balls, while another bag Y contains 3 white balls and 3 black balls. Two balls are drawn (without replacement) at random from one of the bags and were found to be one white and one black. Find the probability that the balls were drawn from bag Y.
Given that E and F are events such that P(E) = 0.6, P(F) = 0.3 and P(E ∩ F) = 0.2, find P (E|F) and P(F|E).
If P(A) = 0.8, P(B) = 0.5 and P(B|A) = 0.4, find P(A ∪ B)
Evaluate P(A ∪ B), if 2P(A) = P(B) = `5/13` and P(A | B) = `2/5`
Determine P(E|F).
A coin is tossed three times, where
E: at least two heads, F: at most two heads
Determine P(E|F).
Two coins are tossed once, where
E: no tail appears, F: no head appears
Determine P(E|F).
Mother, father and son line up at random for a family picture
E: son on one end, F: father in middle
A black and a red dice are rolled.
Find the conditional probability of obtaining the sum 8, given that the red die resulted in a number less than 4.
A fair die is rolled. Consider events E = {1, 3, 5}, F = {2, 3} and G = {2, 3, 4, 5} Find P (E|F) and P (F|E)
Given that the two numbers appearing on throwing the two dice are different. Find the probability of the event ‘the sum of numbers on the dice is 4’.
A and B are two events such that P (A) ≠ 0. Find P (B|A), if A is a subset of B.
A and B are two events such that P (A) ≠ 0. Find P (B|A), if A ∩ B = Φ.
In a game, a man wins a rupee for a six and loses a rupee for any other number when a fair die is thrown. The man decided to throw a die thrice but to quit as and when he gets a six. Find the expected value of the amount he wins/loses.
A card is drawn from a well-shuffled pack of playing cards. What is the probability that it is either a spade or an ace or both?
Five dice are thrown simultaneously. If the occurrence of an odd number in a single dice is considered a success, find the probability of maximum three successes.
Three cards are drawn at random (without replacement) from a well-shuffled pack of 52 playing cards. Find the probability distribution of the number of red cards. Hence, find the mean of the distribution.
In a college, 70% of students pass in Physics, 75% pass in Mathematics and 10% of students fail in both. One student is chosen at random. What is the probability that:
(i) He passes in Physics and Mathematics?
(ii) He passes in Mathematics given that he passes in Physics.
(iii) He passes in Physics given that he passes in Mathematics.
Two dice are thrown simultaneously, If at least one of the dice show a number 5, what is the probability that sum of the numbers on two dice is 9?
An urn contains 4 black, 5 white, and 6 red balls. Two balls are drawn one after the other without replacement, What is the probability that at least one ball is black?
Two cards are drawn one after the other from a pack of 52 cards without replacement. What is the probability that both the cards drawn are face cards?
If P(A) = 0.5, P(B) = 0.8 and P(B/A) = 0.8, find P(A/B) and P(A ∪ B)
Given P(A) = 0.4 and P(A ∪ B) = 0.7 Find P(B) if A and B are independent events
If P(A) = `3/10`, P(B) = `2/5` and P(A ∪ B) = `3/5`, then P(B|A) + P(A|B) equals ______.
Two cards are drawn out randomly from a pack of 52 cards one after the other, without replacement. The probability of first card being a king and second card not being a king is:
Bag I contains 3 red, 4 black and 3 white balls and Bag II contains 2 red, 5 black and 2 white balls. One ball is transferred from Bag I to Bag II and then a ball is draw from Bag II. The ball so drawn is found to be black in colour. Then the probability, that the transferred ball is red, is ______.
If A and B are two events such that `P(A/B) = 2 xx P(B/A)` and P(A) + P(B) = `2/3`, then P(B) is equal to ______.
Read the following passage:
|
Recent studies suggest the roughly 12% of the world population is left-handed.
Assuming that P(A) = P(B) = P(C) = P(D) = `1/4` and L denotes the event that child is left-handed. |
Based on the above information, answer the following questions:
- Find `P(L/C)` (1)
- Find `P(overlineL/A)` (1)
- (a) Find `P(A/L)` (2)
OR
(b) Find the probability that a randomly selected child is left-handed given that exactly one of the parents is left-handed. (2)
A Problem in Mathematics is given to the three students A, B and C. Their chances of solving the problem are `1/2, 1/3` and `1/4` respectively. Find the probability that exactly two students will solve the problem.
If P(B) = `3/5`, P(A | B) = `1/2` and P(A ∪ B) = `4/5`, then P(A ∪ B) + P(A' ∪ B) =
