a) As we are given here that we are selecting the two coins
randomly, therefore:
P(Coin1) = P(Coin2) = 0.5
Also, we are given here that:
P(H | Coin1) = 0.5 and P(H | Coin2) = 1
Using law of total probability, we have here:
P(H) = P(H | Coin1)P(Coin1) + P(H | Coin2) P(Coin2) = 0.5*(0.5 + 1)
= 0.75
a) Given that a heads is the outcome, the probability that we
selected a fair coin is computed using Bayes theorem as:
P(Coin1 | H) = P(H | Coin1)P(Coin1) / P(H) = 0.5*0.5 / 0.75 =
1/3
Therefore 1/3 is the required probability here.
b) Probability that the next toss will result in heads too is
computed here as:
= 0.5*P(Coin1 | H) + 1*P(Coin2 | H)
= 0.5*(1/3) + 1*(2/3)
= 2.5/3 = 5/6
Therefore 5/6 is the required probability here.
c) Given that the netx toss is also heads, probability that the selected coin is fair is computed here as:
P(HH | Coin1) = 0.52 = 0.25,
P(HH | Coin2) = 1
Therefore,
P(HH) = 0.25*0.5 + 1*0.5 = 0.625
Therefore P(Coin1 | HH) = 0.25*0.5 / 0.625 = 0.2
Therefore 0.2 is the required probability here.
Suppose there are two coins. One is a standard fair coin, so that P(heads)=0.50. The other...
Suppose you have two coins. One coin is fair and other is a coin with heads on both sides. Now you choose a coin at random and flip the coin. If the coin lands head, what is the probability that it was the fair coin?
Question 4 (20 points) A box contains three coins, two fair coins and one two-headed coin. (a) You pick a coin at random and toss it. What is the probability that it lands heads up? (b) You pick a coin at random and toss it, and get heads. What is the probability that it is the two-headed coin?
A defective coin minting machine produces coins whose probability of heads is a random variable P with PDF peP, p [0,1], otherwise fp(p) A coin produced by this machine is selected and tossed repeatedly, with successive tosses assumed independent. (a) Find the probability that a coin toss results in heads. (b) Given that a coin toss resulted in heads, find the conditional PDF of P (c) Given that a first coin toss resulted in heads, find the conditional probability of...
You have 2 fair coins and one coin with heads on both sides. You pick a coin at random and toss it twice. If it lands heads up on both tosses, the probability it also lands heads up on a third toss can be express in the form A/B, where A and B are relatively prime positive integers (i.e. the greatest common divisor is 1). Compute A + B.
You have in your pocket two coins, one bent (comes up heads with probability 3/4) and one fair (comes up heads with probability 1/2). Not knowing which is which, you choose one at random and toss it. If it comes up heads you guess that it is the biased coin (reasoning that this is the more likely explanation of the observation), and otherwise you guess it is the fair coin. A) What is the probability that your guess is wrong?
- [10+10]A defective coin minting machine produces coins whose probability of heads is a continuous) random variable P with pdf f(p) = pep ,0<p<1 A coin produced by this machine is selected and tossed. a) Find the probability that the coin toss results in heads. ) Given that the coin toss resulted in heads, find the conditional pdf of P.
A box contains five coins. For each coin there is a different probability that a head will be obtained when the coin is tossed. (Some of the coins are not fair coins!) Let pi denote the probability of a head when the i th coin is tossed (i = 1, . . . , 5), and suppose that p1 = 0, p2 =1/4, p3 =1/2, p4 =3/4, p5 =1. The experiment we are interested in consists in selecting at random...
You have 5 coins, four of which are fair coins, i.e. P(H)=P(T)= 0.5, and the other of which is a two headed coin, i.e. both sides have a head. Suppose you select a coin at random and flip in 3 times, getting all heads. If you flip the coin again, what is the probability it will be heads?
there are two coins. One is fair and the other one has a 5/8 probability to heads. A coin is chosen at random and tossed twice. Heads shows twice. What is the probability the coin you chose is the biased one
a bag contains one fair coin, two two-headed coins, and three two-tailed coins. each of the is flipped, but the outcomes of the fice coins are hidden from you, randomly. if the outcome you see is headsm, what is the probability that the fair coin (which may or may not be the coin that was shown to you) panded heads up?