Advertisements
Advertisements
प्रश्न
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)
Advertisements
उत्तर
Given, P(L) = `12/100`
`P(L) P(overlineL) = 1 - 12/100 = 88/100`
and P(A) = P(B) = P(C) = P(D) = `1/4`
`P(L/A) = 24/100`,
`P(L/B) = 22/100`,
`P(L/C) = 17/100`,
`P(L/D) = 9/100`
(i) `P(L/C) = 17/100`, from the given data.
(ii) `P(overlineL/A) = (P(overlineL ∩ A))/(P(A))`
= `(P(A) - P(L ∩ A))/(P(A))`
= `1 - (P(L ∩ A))/(P(A))`
= `1 - P(L/A)`
= `1 - 24/100`
= `(100 - 24)/100`
= `76/100`
= `38/50`
= `19/25`
(iii) (a) `P(A/L) = (P(A ∩ L))/(P(L))`
But `P(L/A) = (P(A ∩ L))/(P(A))`
`24/100 = (P(A ∩ L))/(1/4)`
`\implies` P(A ∩ L) = `24/100 xx 1/4 = 6/100 = 3/50`
∴ `P(A/L) = (3/50)/(12/100) = (3 xx 100)/(12 xx 50) = 1/2`.
OR
(b) `P(L/(B ∪ C)) = (P[(L) ∩ (B ∪ C)])/(P(B ∪ C))`
= `(P[(L ∩ B) ∪ (L ∩ C)])/(P(B ∪ C))`
= `(P(L ∩ B) + P(L ∩ C) - P(L ∩ B)P(L ∩ C))/(P(B) + P(C) - P(B)P(C))` ...(As they are independent)
= `(22/100 xx 1/4 + 17/100 xx 1/4 - 22/400 xx 17/400)/(1/4 + 1/4 - 1/4 xx 1/4)`
= `(22/400 + 17/400 - (22 xx 17)/(400 xx 400))/(1/2 - 1/16)`
= `((39/400 - 374/160000))/(1/2 - 1/16)`
= `16/7((39 xx 400 - 374)/160000)`
= `(16 xx 15226)/(7 xx 160000)`
= 0.217
= 0.22
APPEARS IN
संबंधित प्रश्न
Assume that each born child 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
- the youngest is a girl.
- at least one is a girl.
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.
An insurance agent insures lives of 5 men, all of the same age and in good health. The probability that a man of this age will survive the next 30 years is known to be 2/3 . Find the probability that in the next 30 years at most 3 men will survive.
Determine P(E|F).
A coin is tossed three times, where
E: head on third toss, F: heads on first two tosses
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 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.
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?
An urn contains 2 white and 2 black balls. A ball is drawn at random. If it is white, it is not replaced into the urn. Otherwise, it is replaced with another ball of the same colour. The process is repeated. Find the probability that the third ball is drawn is black.
Two balls are drawn from an urn containing 3 white, 5 red and 2 black balls, one by one without replacement. What is the probability that at least one ball is red?
Bag A contains 4 white balls and 3 black balls. While Bag B contains 3 white balls and 5 black balls. Two balls are drawn from Bag A and placed in Bag B. Then, what is the probability of drawing a white ball from Bag B?
In an examination, 30% of students have failed in subject I, 20% of the students have failed in subject II and 10% have failed in both subject I and subject II. A student is selected at random, what is the probability that the student has failed in exactly one subject?
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 balls are drawn from an urn containing 5 green, 3 blue, and 7 yellow balls one by one without replacement. What is the probability that at least one ball is blue?
Three fair coins are tossed. What is the probability of getting three heads given that at least two coins show heads?
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 for two events A and B, P(A) = `3/4`, P(B) = `2/5` and A ∪ B = S (sample space), find the conditional probability P(A/B)
A problem in Mathematics is given to three students whose chances of solving it are `1/3, 1/4` and `1/5`. What is the probability that exactly one of them will solve it?
One bag contains 5 white and 3 black balls. Another bag contains 4 white and 6 black balls. If one ball is drawn from each bag, find the probability that both are white
Two thirds of students in a class are boys and rest girls. It is known that the probability of a girl getting a first grade is 0.85 and that of boys is 0.70. Find the probability that a student chosen at random will get first grade marks.
Given P(A) = 0.4 and P(A ∪ B) = 0.7 Find P(B) if P(A/B) = 0.4
A year is selected at random. What is the probability that it contains 53 Sundays
If P(A) = `2/5`, P(B) = `3/10` and P(A ∩ B) = `1/5`, then P(A|B).P(B'|A') is equal to ______.
If for two events A and B, P(A – B) = `1/5` and P(A) = `3/5`, then `P(B/A)` is equal to ______.
Students of under graduation submitted a case study on “Understanding the Probability of Left-Handedness in Children Based on Parental Handedness”. Following Recent studies suggest that roughly 12% of the world population is left-handed. Depending on the parents’ handedness, the chances of having a left-handed child are as follows:
Scenario A: Both parents are left-handed, with a 24% chance of the child being left-handed.
Scenario B: The fathers is right-handed and the mothers left-handed, with a 22% chance of child being left-handed.
Scenario C: The fathers left-handed and the mother is right-handed, with a 17% chance of child being left-handed.
Scenario D: Both parents are right-handed, with a 9% chance of having a left-handed child.
Assuming that scenarios A, B, C and D are equally likely and L denotes the event that the child is left-handed, answer the following questions.
- What is the overall probability that a randomly selected child is left-handed?
- Given that exactly one parent is left-handed, what is the probability that a randomly selected child is left-handed?
- If a child is left-handed, what is the probability that both parents are left-handed?
Which conditional probabilities are equal to 1 when \[P(F)\neq 0\]?
