Q1. There are 26 letters in the English alphabet; A, E, I, O, and U are vowels; the rest are consonants. A box contains the letters O, R, A, N, G, E, S. Three letters are drawn at random with replacement. Find the chance that
a) all the letters drawn are consonants
b) not all the letters drawn are vowels
c) at least one of the letters drawn is a vowel
Q2. Three draws are made at random without replacement from a box which contains 4 red tickets, 4 blue tickets, and 2 green tickets. The conditional chance that all three draws are the same color, given that the first two are red, is
(i) (4/10)x(3/9)x(2/8)
(ii) 2/8
(iii) other (specify): ______________
Q3. A box contains the letters O, R, A, N, G, E, S. Letters are drawn one by one at random without replacement until the box is empty. Find the chance that
a) the letter R appears on the last draw: _____________
b) the letter R appears on the last draw, given that a vowel appeared on the first draw:______________
c) vowels appear on the first two draws: ______________
d) there is one vowel and one consonant among the first two letters drawn: _____________
Q4. In an undergraduate class, 70% of the students are juniors and the rest are seniors. Of the juniors, 40% are math majors. Of the seniors, 25% are math majors. One student is picked at random. Find
a) P(junior) b) P(senior) c) P(math major | junior)
d) P(not a math major | junior) e) P(math major | senior)
f) P(not a math major | senior) g) P(junior math major)
h) P(senior math major) i) P(math major)
j) P(junior | math major) k) P(senior | math major)
l) P(junior | not a math major)
Q5. An electrical firm manufactures light bulbs that have a life, before burn-out, that is normally distributed with mean equal to 800 hours and a standard deviation of 40 hours. Find the probability that a bulb burns between 778 and 834 hours.
Q6. Suppose that 10% of a given population has benign chronic flatulence. Suppose that there is a standard screening test for benign chronic flatulence that has a 90% chance of correctly detecting that one has the disease, and a 10% chance of a false positive (erroneously reporting that one has the disease when one does not). We pick a person at random from the population (so that everyone has the same chance of being picked) and test him/her. The test is positive. What is the chance that the person has the disease?