Advertisements
Advertisements
प्रश्न
Bag A contains 3 red and 2 white balls and bag B contains 2 red and 5 white balls. A bag is selected at random, a ball is drawn and put into the other bag, and then a ball is drawn from that bag. Find the probability that both the balls drawn are of same color
Advertisements
उत्तर
Let event C1: First ball drawn is red and from bag A,
event D1: First ball drawn is white and from bag A,
event E1: First ball drawn is red and from bag B,
event F1: First ball drawn is white and from bag B,
event C2: Second ball drawn is red and from bag B,
event D2: Second ball drawn is white and from bag B,
event E2: Second ball drawn is red and from bag A,
event F2: Second ball drawn is white and from bag A,
event G: Selecting bag A in first place,
event H: Selecting bag B in first place.
P(G) = P(H) = `1/2`
Let event X: Both the balls drawn are of same color.
∴ P(X) = `"P"("G") xx "P"("X"//"G") + "P"("H") xx "P"("X"//"H")` …(i)
If bag A is selected in first place, then In bag A, we have 5 balls, out of which 3 are red.
∴ Probability of getting first red ball from bag A = P(C1)
= `(""^3"C"_1)/(""^5"C"_1) = 3/5`
Since first red ball is put into the bag B, we now have 8 balls in bag B, out of which 3 are red.
∴ Probability of getting second red ball from bag B.
`"P"("C"_2//"C"_1) = (""^3"C"_1)/(""^8"C"_1) = 3/8`
Similarly, probability of getting first white ball from bag A = P(D1) = `(""^2"C"_1)/(""^5"C"_1) = 2/5` and probability of getting second white ball form bag B = `"P"("D"_2//"D"_1) = (""^6"C"_1)/(""^8"C"_1) = 6/8`
∴ `"P"("X"//"G") = "P"("C"_1) * "P"("C"_2//"C"_1) + "P"("D"_1) * "P"("D"_2//"D"_1)`
= `3/5 xx 3/8 + 2/5 xx 6/8`
= `21/40` ...(ii)
Similarly, `"P"("X"//"H")`
= `"P"("E"_1) * "P"("E"_2//"E"_1) + "P"("F"_1) * "P"("F"_2//"F"_1)`
= `2/7 xx 4/6 + 5/7 xx 3/6`
= `23/42` ...(iii)
From (i), (ii), (iii),
Required probability = `1/2 xx 21/40 + 1/2 xx 23/42`
= `3604/6720`
= `901/1680`
APPEARS IN
संबंधित प्रश्न
If `P(A) = 3/5 and P(B) = 1/5` , find P (A ∩ B) if A and B are independent events.
A fair coin and an unbiased die are tossed. Let A be the event ‘head appears on the coin’ and B be the event ‘3 on the die’. Check whether A and B are independent events or not.
Let E and F be events with `P(E) = 3/5, P(F) = 3/10 and P(E ∩ F) = 1/5`. Are E and F independent?
Two events, A and B, will be independent if ______.
Prove that if E and F are independent events, then the events E and F' are also independent.
In a race, the probabilities of A and B winning the race are `1/3` and `1/6` respectively. Find the probability of neither of them winning the race.
The probability that a 50-year old man will be alive till age 60 is 0.83 and the probability that a 45-year old woman will be alive till age 55 is 0.97. What is the probability that a man whose age is 50 and his wife whose age is 45 will both be alive after 10 years?
An urn contains four tickets marked with numbers 112, 121, 122, 222 and one ticket is drawn at random. Let Ai (i = 1, 2, 3) be the event that ith digit of the number of the ticket drawn is 1. Discuss the independence of the events A1, A2, and A3.
A, B, and C try to hit a target simultaneously but independently. Their respective probabilities of hitting the target are `3/4, 1/2` and `5/8`. Find the probability that the target
- is hit exactly by one of them
- is not hit by any one of them
- is hit
- is exactly hit by two of them
Two dice are thrown together. Let A be the event 'getting 6 on the first die' and B be the event 'getting 2 on the second die'. Are the events A and B independent?
The probability that a man who is 45 years old will be alive till he becomes 70 is `5/12`. The probability that his wife who is 40 years old will be alive till she becomes 65 is `3/8`. What is the probability that, 25 years hence,
- the couple will be alive
- exactly one of them will be alive
- none of them will be alive
- at least one of them will be alive
Select the correct option from the given alternatives :
The odds against an event are 5:3 and the odds in favour of another independent event are 7:5. The probability that at least one of the two events will occur is
Solve the following:
Let A and B be independent events with P(A) = `1/4`, and P(A ∪ B) = 2P(B) – P(A). Find `"P"("A"/"B")`
Solve the following:
Find the probability that a year selected will have 53 Wednesdays
Solve the following:
Consider independent trails consisting of rolling a pair of fair dice, over and over What is the probability that a sum of 5 appears before sum of 7?
Let A and B be two independent events. Then P(A ∩ B) = P(A) + P(B)
Three events A, B and C are said to be independent if P(A ∩ B ∩ C) = P(A) P(B) P(C).
For a loaded die, the probabilities of outcomes are given as under:
P(1) = P(2) = 0.2, P(3) = P(5) = P(6) = 0.1 and P(4) = 0.3. The die is thrown two times. Let A and B be the events, ‘same number each time’, and ‘a total score is 10 or more’, respectively. Determine whether or not A and B are independent.
Refer to Question 1 above. If the die were fair, determine whether or not the events A and B are independent.
In Question 64 above, P(B|A′) is equal to ______.
If A and B are such events that P(A) > 0 and P(B) ≠ 1, then P(A′|B′) equals ______.
If A and B are independent events, then A′ and B′ are also independent
If A and B′ are independent events, then P(A' ∪ B) = 1 – P (A) P(B')
If A and B are independent, then P(exactly one of A, B occurs) = P(A)P(B') + P(B)P(A')
One card is drawn at random from a well-shuffled deck of 52 cards. In which of the following case is the events E and F independent?
E : ‘the card drawn is black’
F : ‘the card drawn is a king’
If A, B are two events such that `1/8 ≤ P(A ∩ B) ≤ 3/8` then
If P(A) = `3/5` and P(B) = `1/5`, find P(A ∩ B), If A and B are independent events.
Events A and Bare such that P(A) = `1/2`, P(B) = `7/12` and `P(barA ∪ barB) = 1/4`. Find whether the events A and B are independent or not.
Five fair coins are tossed simultaneously. The probability of the events that at least one head comes up is ______.
A problem in Mathematics is given to three students whose chances of solving it are `1/2, 1/3, 1/4` respectively. If the events of their solving the problem are independent then the probability that the problem will be solved, is ______.
Given two events A and B such that (A/B) = 0.25 and P(A ∩ B) = 0.12. The value P(A ∩ B') is ______.
