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The probability that any child in a certain family will have blue eyes is 1/4, and this feature is inherited independently by different children in the family. If there are five children in the family and it is known that at least one of these children has blue eyes, what is the probability that at least three of the children have blue eyes?

Suppose that a balanced die is rolled three times, and let\({X_i}\)denote the number that appears on the ith roll (i = 1, 2, 3). Evaluate\({\rm P}\left( {{X_1} > {X_2} > X3} \right)\).

Suppose that a box contains one fair coin and one coin with a head on each side. Suppose that a coin is drawn at random from this box and that we begin to flip the coin. In Eqs. (2.3.4) and (2.3.5), we computed the conditional probability that the coin was fair, given that the first two flips both produce heads.

a. Suppose that the coin is flipped a third time and another head is obtained. Compute the probability that the coin is fair, given that all three flips produced

heads.

b. Suppose that the coin is flipped a fourth time, and the result is tails. Compute the posterior probability that the coin is fair.

If five balls are thrown at random into n boxes, and all throws are independent, what is the probability that no box contains more than two balls?

If A and B are independent events and Pr(B) < 1, what is the value of Pr(A^c|B^c)?

If A⊂B with Pr(B) > 0, what is the value of \({\bf{Pr}}\left( {{\bf{A}}|{\bf{B}}} \right)\) ?

Suppose that two players A and B take turns rolling a pair of balanced dice and that the winner is the first player who obtains the sum of 7 on a given roll of the two dice. If A rolls first, what is the probability that B will win?

Suppose that 80 percent of all statisticians are shy, whereas only 15 percent of all economists are shy. Suppose also that 90 percent of the people at a large gathering are economists and the other 10 percent are statisticians. If you meet a shy person at random at the gathering, what is the probability that the person is a statistician?

Dreamboat cars are produced at three different factories A, B, and C. Factory A produces 20 percent of the total output of Dreamboats, B produces 50 percent, and C produces 30 percent. However, 5 percent of the cars produced at A are lemons, 2 percent of those produced at B are lemons, and 10 percent of those produced at C are lemons. If you buy a Dreamboat and it turns out to be a lemon, what is the probability that it was produced at factory A?

Suppose that 30 percent of the bottles produced in a certain plant are defective. If a bottle is defective, the probability is 0.9 that an inspector will notice it and remove it from the filling line. If a bottle is not defective, the probability is 0.2 that the inspector will think that it is defective and remove it from the filling line.

a. If a bottle is removed from the filling line, what is the probability that it is defective?

b. If a customer buys a bottle that has not been removed from the filling line, what is the probability that it is defective?