University of the West Indies
COMP4610 - Statistics for Data Science
Assignment 1.
Due date: September 22, 2021.
1. In a large group of people, it is known that 10% plays cricket, 20% plays football and 25%
plays cricket or football. Find the probability that a person chosen at random from this group
(a) plays cricket and plays football.
[1]
(b) plays football, given that the person chosen plays cricket
[2]
2. Two ordinary dice are thrown. Find the probability that
(a) at least one six is thrown;
[1]
(b) at least one two is thrown;
[1]
(c) at least one six or at least one two is thrown.
[2]
3. When a person needs a taxi, it is hired from one of the three firms, X, Y and Z. Of the hirings:
40% are from X, 50% are from Y and 10% are from Z. For taxis hired from X, 9% arrive late,
the corresponding percentages for taxis hired from firms Y and Z that arrive late are 6% and
20% respectively. Calculate the probability that the next taxi hired:
(a) will be from firm X and will not be late;
[2]
(b) will arrive late.
[3]
4. I have two boxes. The first contains 5 white balls, 4 blue balls, and 2 red balls. The second
contains 2 white balls, 2 blue balls and 1 red ball. Without looking, I draw one ball at random
from the first box and transfer it to the second box. Afterwards I draw a second ball at random
from the second box.
(a) What is the probability that the second ball is red?
[3]
(b) What is the probability that the first ball is blue if the second one is red?
[3]
5. For the events A and B: P(A) = 0.7, P(A ∪ B) = 0.9 and P(A ∩ B) = 0.3. Find the following
(a) P(B);
[1]
(b) P(B 0 ∩ A);
[2]
(c) P(B ∩ A0 );
[2]
(d) P(B 0 ∩ A0 ).
[2]
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6. A batch of twenty items taken from a production line, contains three which are faulty.
An inspector takes two at random from the twenty items. What is the probability that:
(a) both items are faulty;
[2]
(b) at least one is faulty.
[2]
7. A poker hand has five cards drawn from an ordinary deck of 52 cards.
Find the probability that the poker hand has exactly 2 kings.
[2]
8. If you hold 3 tickets to a lottery for which n tickets were sold and 5 prizes are to be
given, what is the probability that you will win at least 1 prize?
[5]
9. Two boxes each have r balls labelled 1, 2, . . . , r. A random sample of size n ≤ r is
drawn without replacement from each box. Find the probability that the samples
contain exactly k balls having the same numbers in common.
[5]
10. Prizes are awarded to four different members of a group of eight people.
Find the number of ways in which the prizes are awarded
(a) if there is a 1st prize, a 2nd prize, a 3rd prize and a 4th prize,
[2]
(b) if there are two 1st prizes and two 2nd prizes.
[2]
11. (a) A panel of judges in an essay competition has to select, and place in order of merit,
4 winning entries from a total of 20. Find the number of ways in which this can be
done.
(b) As first step in the selection, 5 finalists are selected, without being placed in order.
Find the number of ways in which this can be done.
[1]
[2]
(c) All 20 essays are subsequently assessed by the panels of judges for content, accuracy
and style, respectively, and three special prizes are awarded, one by each panel.
Find the number of ways in which this can be done, assuming that an essay may win
more than one prize.
[2]
END OF ASSIGNMENT
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