Conditional Probability 1. A grade school boy has five blue and four whites marbles in his left pocket and four blue and five white marbles in his right pocket. If he transfers one marble at random from his left to his right pocket, what is the probability of then drawing a blue marble from his right pocket? 2. Suppose that the genes for eye color for a certain male fruit fly are (R.W) and the genes for eye color for the mating female fruit fly are (R.W), where R and W represent red and white respectively. Their offspring receive one gene for eye color from each parent. Define the sample space for the genes for eye color for the offspring. Assume that each of the four possible outcomes has equal probability. If an offspring ends up with two red genes or one red and one white gene for eye color, its eyes will look red. Given that an offspring's eyes look red, what is the conditional probability that it has two red genes for eye color? 3. In the gambling game "craps" a pair of dice is rolled and the outcome of the experiment is the sum of the dice. The bettor wins on the first roll if the sum is 7 or 11. The bettor loses on the first roll if the sum is 2, 3, or 12. If the sum is 4, 5, 6, 8, 9, or 10, that number is called the bettor's "point". Once the "point" is established, the rule is: If the bettor rolls a 7 before the "point", the bettor loses; but if the "point" is rolled before the 7, the bettor wins. List the 36 outcomes in the sample space for the roll of a pair of dice. Assume each of them has a probability of 1/36. Find the probability that the bettor wins on the first roll. That is, find the probability of rolling a 7 or 11, P(7 or 11). Given that 8 is the outcome on the first roll, find the probability that the bettor now rolls the point 8 before rolling a 7 and thus wins. Note that at this stage of the game the only outcomes of interest are 7 and 8. Thus find P(8|7 or 8). The probability that a bettor rolls an 8 on the first roll and then wins is given by P(8|7 or 8). Show that the value of this is (5/36)(5/11). Show that the total probability that a bettor wins in the game of craps is 0.49293. 4. Consider the birthdays of the students in a class of size r. Assume the year consist of 365 days. How many different ordered samples of birthdays are possible (r in sample) allowing repetitions (with replacements)? The same as (a) except requiring that all students have different birthdays (w/out replacement)? If we can assume that each ordered outcome in part (a) has the same probability, what is the probability that no two students have the same birthday? For what value of r is the probability in part (c) about equal to 12? Is this surprisingly small? Hint: use a calculator to find r