Question
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Ten runners are in a race in which first second and third place will be awarded. Assuming that there are no ties, how many different outcomes are possible? Thirty photographs are entered in a photo competition. How many ways are there to award the top five places? 3. Two math students and three computer science students come to class late. There are five students in total. How many ways can the five students walk in through the door late (i. e. 1st, 2nd: 3rd etc)? How many ways can the students come in, if a computer science student must be first? How many subsets with exactly 2 elements does a set with 8 elements have? How- many subsets with an odd number of elements does a set with 8 elements have? How- many subsets with at most 2 elements does a set with 100 elements have? Suppose a die with six sides (1,2,3,4,5,6) is tossed. What is the probability that the number that appears is a 2? What is the probability that the number that appears is divisible by 3?(c) If the die is tossed twice, find the probability that the sum of the numbers appearing is at least 6. Suppose a fair coin is tossed 4 times. Write out all possible outcomes (i.e. HHTT: etc). What is the probability of obtaining four tails? What is the probability of obtaining two heads and two tails? What is the probability of no heads? What is the probability of obtaining at least 1 head? A bag contains 4 blue marbles, 2 green marbles, 1 white marble and 5 red marbles. Suppose a marble is selected from the bag and replaced. What is the probability of receiving a blue marble on the first selection and a blue marble on the second selection? Suppose a marble is selected from the bag and replaced. What is the probability of receiving a blue marble on the first selection and not a blue marble on the second selection? Suppose a marble is selected from the bag and replaced. What is the probability of receiving at least one blue marble and a white marble (in either order)? Suppose a marble is selected from the bag and not replaced. What is the probability of receiving a blue marble on the first selection and a blue marble on the second selection?
Explanation / Answer
1. 10 * 9 * 8 = 720
2. 30 * 29 * 28 * 27 * 26 = 17100720
3. (a) 5 * 4* 3* 2 * 1 = 120
(b) 3 * 4* 3 * 2 * 1 = 72
4. 8 choose 2 = 8 * 7 / 2 = 28
5. 8 choose 1 + 8 choose 3 + 8 choose 5 + 8 choose 7 = 8 + 56 + 56 + 8 = 128
6. 100 choose 0 + 100 choose 1 + 100 choose 2 = 1 + 100 + 100*99/2 = 1 + 100 + 4950 = 5051
7. (a) 1/6
(b) 3 and 6 are divisible by 3, so the probability is 2/6 = 1/3
(c) P(X + Y >= 6) = P(X=1)*P(Y >=5) + P(X = 2)*P(Y>=4) + P(X=3)*P(Y >=3) + P(X = 4)*P(Y>=2)
+ P(X=5)*P(Y >=1) + P(X = 6)*P(Y>=0)
= 1/6 * 2/6 + 1/6 * 3/6 + 1/6 * 4/6 + 1/6 * 5/6 + 1/6 * 6/6 + 1/6 * 6/6 = (2 + 3+ 4+ 5+ 6 + 6)/36 = 26/36 = 13/18
8. (a)
16 outcomes
HHHH, HHHT, HHTH, HHTT, HTHH, HTHT, HTTH, HTTT,
THHH, THHT, THTH, THTT, TTHH, TTHT, TTTH, TTTT
(b)1/16
(c) (4 choose 2) (1/2)^2 (1/2)^2 = 6/16 = 3/8
(d) 1/16
(e) 1 - 1/16 = 15/16
9.
(a) There are 12 total marbles. With replacement, the probability of picking 2 blue marbles is
(4/12)* (4/12) = (1/3)(1/3) = (1/9)
(b) The probability is
(4/12)*(8/12) = 1/3 * 2/3 = 2/9
(c) P(BW) + P(WB) = 4/12 * 1/12 + 1/12 * 4/12 = 8/144 = 1/18
(d) Now without replacement, if we pick a blue marble first, the chance we pick a second one is 3/11. So the probability is
4/12 * 3/11 = 1/11