Hi all,
This problem is an example in the book "Introduction To Probability", and I am having trouble understanding a specific part in it. The problems is as follows: two design teams called A and B are given a design task; the probability of success for A is 2/3, and for B it's 1/2. The probability that at least one team will succeed is 3/4. If both teams are successful, the design of B is adopted, assuming that exactly 1 successful design is produced, what is the probability that is was designed by team N?
The given solution in the book: There are four possible outcomes - SS, SF, FS, and FF (the first letter corresponds to team A, the latter for team B; S stands for success, F stands for failure).
The given probabilities are as follows(SS) + P(SF) = 2/3, P(SS) + P(FS) = 1/2, P(SS) + P(SF) + P(FS) = 3/4
From these relations, together with the normalization equation P(SS) + P(SF) + P(FS) + P(FF) = 1, we can obtain the probabilities of all the outcomes.
This is the part I can't figure out. How are the above relations used to work out the probabilites for the individual events? The book goes on to specify that P(SS) = 5/12, P(SF) = 1/4, P(FS) = 1/12, P(FF) = 1/4. I'm not really sure how these were worked out, having spent quite some time trying to work it out but to no avail. Any thoughts or help would be most appreciated.
This problem is an example in the book "Introduction To Probability", and I am having trouble understanding a specific part in it. The problems is as follows: two design teams called A and B are given a design task; the probability of success for A is 2/3, and for B it's 1/2. The probability that at least one team will succeed is 3/4. If both teams are successful, the design of B is adopted, assuming that exactly 1 successful design is produced, what is the probability that is was designed by team N?
The given solution in the book: There are four possible outcomes - SS, SF, FS, and FF (the first letter corresponds to team A, the latter for team B; S stands for success, F stands for failure).
The given probabilities are as follows(SS) + P(SF) = 2/3, P(SS) + P(FS) = 1/2, P(SS) + P(SF) + P(FS) = 3/4
From these relations, together with the normalization equation P(SS) + P(SF) + P(FS) + P(FF) = 1, we can obtain the probabilities of all the outcomes.
This is the part I can't figure out. How are the above relations used to work out the probabilites for the individual events? The book goes on to specify that P(SS) = 5/12, P(SF) = 1/4, P(FS) = 1/12, P(FF) = 1/4. I'm not really sure how these were worked out, having spent quite some time trying to work it out but to no avail. Any thoughts or help would be most appreciated.