SIT1001 Probability and Statistics I Tutorial 1
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SIT1001 Probability and Statistics I
Tutorial 1
1. Describe the outcome space for each of the following experiments:
a. A candy bar with a 20.4-gram label weight is selected at random from a production line and is weighed.
b. A coin is tossed three times, and the sequence of heads and tails is observed.
2. The National Geographic reported the following litter sizes for 20 lions: 2, 2, 1, 3, 2, 2, 3, 1, 4, 3, 2, 1, 3, 2, 2, 2, 2, 1, 2, 1.
a. Construct a frequency table for these data.
b. Draw a frequency histogram.
c. Draw a relative frequency histogram.
d. Is there a typical litter size?
3. Toss two coins at random and count the number of heads that appear “up”. Here
��� = {0,1,2}. A reasonable probability model is given by the p.m.f. ���(0) = 1/4,
���(1) = 1/2, and ���(2) = 1/4. Repeat this experiment at least ��� = 100 times and plot the resulting relative frequency histogram ℎ(���) on the same graph with ���(���).
4. A fair eight-sided die is rolled once. Let ��� = {2,4,6,8}, ��� = {3,6}, ��� = {2,5,7}
and ��� = {1,3,5,7}. Assume that each face has the same probability.
a. Give the values of (i) ���(���), (ii) ���(���), (iii) ���(���) and (iv) ���(���).
[4/8,2/8,3/8,4/8]
b. Give the values of (i) ���(��� ∩ ���), (ii) ���(��� ∩ ���) and (iii) ���(��� ∩ ���).
[1/8,0,2/8]
c. Give the values of (i) ���(��� ∪ ���), (ii) ���(��� ∪ ���) and (iii) ���(��� ∪ ���).
[5/8,5/8,5/8]
5. A typical roulette wheel used in a casino has 38 slots that are numbered 1,2,..,36,0,00, respectively. The 0 and 00 slots are colored green. Half of the remaining slots are red and half are black. Also half of the integers between 1 and 36 inclusive are odd, half are even, and 0 and 00 are defined to be neither odd nor even. A ball is rolled around the wheel and ends up in one of the slots; we assume each slot has equal probability of 1/38 and we are interested in the number of the slot in which the ball falls.
a. Define the sample space ���.
b. Let ��� = {0,00}. Give the value of ���(���).
c. Let ��� = {14,15,17,18}. Give the value of ���(���′).
d. Let ��� = {���: ��� is odd}. Give the value of ���(���).
[��� = {00,0,1,2, … ,36}, 2 , 34 , 18]
38 38 38
6. Divide a line segment into two parts by selecting a point at random. Use your intuition to assign a probability to the event that the larger segment is at least two times longer than the shorter segment.
[2/3]
7. A boy found a bicycle lock for which the combination was unknown. The correct combination is a four-digit number, ���1���2���3���4, where ������, ��� = 1,2,3,4, is selected
from 1,2,3,4,5,6,7,8. How many different lock combinations are possible with such a lock? [4096]
8. How many four-letter code words are possible using the letters in HOPE if
a. The letters may not be repeated?
b. The letters may be repeated?
[24,256]
9. In a state lottery four digits are drawn at random one at a time with replacement from 0 to 9. Suppose that you win if any permutation of your selected integers is drawn. Give the probability of winning if you select
a. 6,7,8,9
b. 6,7,8,8
c. 7,7,8,8
d. 7,8,8,8
[0.0024,0.0012,0.0006,0.0004]
10. Pascal’s triangle give a method for calculating the binomial coefficients; it begins as follows:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
⋮ ⋮ ⋮ ⋮ ⋮
The nth row of this triangle gives the coefficients for (��� + ���)���−1. To find an entry in the table other than a 1 on the boundary, add the two nearest numbers in the row directly above. The equation
��� ��� − 1 ��� − 1
(���) = ( ��� ) + (��� − 1)
is called the Pascal’s equation. Explain why Pascal’s triangle works. Prove that
the equation is correct.
11. A poker hand is defined as drawing five cards at random without replacement from a deck of 52 playing cards. Find the probability of each of the following poker hands:
a. Four of a kind (four cards of equal face value and one card of a different value).
b. Full house (one pair and one triple of cards with equal face value).
c. Three of a kind (three equal face values plus two cards of different values).
d. Two pairs (two pairs of equal face value plus one card of a different value).
e. One pair (one pair of equal face value plus three cards of different values).
[0.00024,0.00144,0.02113,0.04754,0.42257]
12. There are three teams in a cross country race. Team A has 5 runners, team B has 6 runners and team C has 7 runners. In how many ways can the runners cross the finish line if we are only interested in the team for which they run? That is, what is the number of distinguishable permutations of 5 A’s, 6 B’s and 7 C’s? (Note that for scoring purposes, only the scores of the first 5 runners for each team count.) [14702688]
2022-11-23