Cards are drawn one by one at random from a well-shuffled full pack of 52 playing cards until 2 aces are obtained for the first time. If N is the number of cards required to be drawn, then show that Pr{N=n}=50×49×17×13(n−1)(52−n)(51−n) where 2≤n≤50.
Suppose the probability for A to win a game against B is 0.4. If A has an option of playing either a "best of 3 games" or a "best of 5 games" match against B, which option should be choose so that the probability of his winning the match is higher? (No game ends in a draw).
A lot contains 50 defective and 50 non defective bulbs. Two bulbs are drawn at random, one at a time, with replacement. The events A,B,C are defined as A=(the first bulb is defective), B=(the second bulb is non-defective), C=(the two bulbs are both defective or both non defective). Determine whether (i) A,B,C are pairwise independent (ii) A,B,C are independent.
In how many ways three girls and nine boys can be seated in two vans, each having numbered seats, 3 in the front and 4 at the back? How many seating arrangements are possible if 3 girls should sit together in a back row on adjacent seats? Now, if all the seating arrangements are equally likely, what is the probability of 3 girls sitting together in a back row on adjacent seats?
Three players, A,B and C, toss a coin cyclically in that order (that is A,B,C,A,B,C,A,B,…) till a head shows. Let p be the probability that the coin shows a head. Let α,β and γ be, respectively, the probabilities that A,B and C gets the first head. Prove that β=(1−p)α. Determine α,β and γ (in terms of p).
A coin has probability p of showing head when tossed. It is tossed n times. Let pn denote the probability that no two (or more) consecutive heads occur. Prove that p1=1,p2=1−p2 and pn=(1−p)pn−1+p(1−p)pn−2 for all n≥3.
A coin has probability p of showing head when tossed. It is tossed n times. Let pn denote the probability that no two (or more) consecutive heads occur. Prove that p1=1,p2=1−p2 and pn=(1−p)pn−1+p(1−p)pn−2 for all n≥3. Prove by induction on n, that pn=Aαn+Bβn for all n≥1, where α and β are the roots of quadratic equation x2−(1−p)x−p(1−p)=0 and A=αβ−α2p2+β−1,B=αβ−β2p2+α−1.
A and B are two independent events. C is event in which exactly one of A or B occurs. Prove that P(C)≥P(A∪B)P(Aˉ∩Bˉ).
Balls are drawn one-by-one without replacement from a box containing 2 black, 4 white and 3 red balls till all the balls are drawn. Find the probability that the balls drawn are in the order 2 black, 4 white and 3 red.
For a biased die the probabilities for the different faces to turn up are given below : FacePr ob.10.120.3230.2140.1550.0560.17 This die is tossed and you are told that either face 1 or face 2 has turned up. Then the probability that it is face 1 is .........