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
Let A and B be independent events with P (A) = 0.3 and P (B) = 0.4. Find
- P (A ∩ B)
- P (A ∪ B)
- P (A | B)
- P (B | A)
Events A and B are such that `P(A) = 1/2, P(B) = 7/12 and P("not A or not B") = 1/4` . State whether A and B are independent?
If each element of a second order determinant is either zero or one, what is the probability that the value of the determinant is positive? (Assume that the individual entries of the determinant are chosen independently, each value being assumed with probability `1/2`).
Prove that if E and F are independent events, then the events E and F' are also independent.
The probabilities of solving a specific problem independently by A and B are `1/3` and `1/5` respectively. If both try to solve the problem independently, find the probability that the problem is solved.
A problem in statistics is given to three students A, B, and C. Their chances of solving the problem are `1/3`, `1/4`, and `1/5` respectively. If all of them try independently, what is the probability that, problem is solved?
A problem in statistics is given to three students A, B, and C. Their chances of solving the problem are `1/3`, `1/4`, and `1/5` respectively. If all of them try independently, what is the probability that, problem is not solved
The odds against a husband who is 55 years old living till he is 75 is 8: 5 and it is 4: 3 against his wife who is now 48, living till she is 68. Find the probability that the couple will be alive 20 years hence.
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?
A bag contains 3 yellow and 5 brown balls. Another bag contains 4 yellow and 6 brown balls. If one ball is drawn from each bag, what is the probability that, both the balls are of the same color?
A bag contains 3 red and 5 white balls. Two balls are drawn at random one after the other without replacement. Find the probability that both the balls are white.
Solution: Let,
A : First ball drawn is white
B : second ball drawn in white.
P(A) = `square/square`
After drawing the first ball, without replacing it into the bag a second ball is drawn from the remaining `square` balls.
∴ P(B/A) = `square/square`
∴ P(Both balls are white) = P(A ∩ B)
`= "P"(square) * "P"(square)`
`= square * square`
= `square`
A family has two children. Find the probability that both the children are girls, given that atleast one of them is a girl.
Solve the following:
If P(A) = `"P"("A"/"B") = 1/5, "P"("B"/"A") = 1/3` the find `"P"("B'"/"A'")`
Solve the following:
A machine produces parts that are either good (90%), slightly defective (2%), or obviously defective (8%). Produced parts get passed through an automatic inspection machine, which is able to detect any part that is obviously defective and discard it. What is the quality of the parts that make it throught the inspection machine and get shipped?
If A and B are independent events such that 0 < P(A) < 1 and 0 < P(B) < 1, then which of the following is not correct?
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.
A and B are two events such that P(A) = `1/2`, P(B) = `1/3` and P(A ∩ B) = `1/4`. Find: `"P"("A"/"B")`
Let E1 and E2 be two independent events such that P(E1) = P1 and P(E2) = P2. Describe in words of the events whose probabilities are: P1P2
Let E1 and E2 be two independent events such that P(E1) = P1 and P(E2) = P2. Describe in words of the events whose probabilities are: P1 + P2 – 2P1P2
Two dice are tossed. Find whether the following two events A and B are independent: A = {(x, y): x + y = 11} B = {(x, y): x ≠ 5} where (x, y) denotes a typical sample point.
If A and B are two events such that P(B) = `3/5`, P(A|B) = `1/2` and P(A ∪ B) = `4/5`, then P(A) equals ______.
In Question 64 above, P(B|A′) is equal to ______.
If A and B are two independent events with P(A) = `3/5` and P(B) = `4/9`, then P(A′ ∩ B′) equals ______.
If two events are independent, then ______.
Two independent events are always mutually exclusive.
If A and B′ are independent events, then P(A' ∪ B) = 1 – P (A) P(B')
If A and B are two events such that P(A) > 0 and P(A) + P(B) >1, then P(B|A) ≥ `1 - ("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’
Let E1 and E2 be two independent events. Let P(E) denotes the probability of the occurrence of the event E. Further, let E'1 and E'2 denote the complements of E1 and E2, respectively. If P(E'1 ∩ E2) = `2/15` and P(E1 ∩ E'2) = `1/6`, then P(E1) is
The probability that A hits the target is `1/3` and the probability that B hits it, is `2/5`. If both try to hit the target independently, find the probability that the target is hit.
Let EC denote the complement of an event E. Let E1, E2 and E3 be any pairwise independent events with P(E1) > 0 and P(E1 ∩ E2 ∩ E3) = 0. Then `"P"(("E"_2^"C" ∩ "E"_3^"C")/"E"_1)` is equal to ______.
Two players A and B are alternately throwing a coin and a die together. A player who first throws head and 6 wins the game. If A starts the game, then the probability that B wins the game is ______.
Let E and F be two independent events. The probability that both E and F happen is `1/12` and the probability that neither E nor F happens is `1/2`, then a value of `(P(E))/(P(F))` is ______.
What are independent events in probability?
Which is the equivalent and most commonly used test for independent events E and F?
A die is thrown once. If F is the event that the number appearing is even, which set represents F?
For \[E\cap F=\{6\}\] when a die is thrown once, what is \[P(E\cap F)\]?
