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Six Probability Problems

1) In a survey of 125 college students, it was found that of three newspapers, the Wall Street Journal, New York Times, and Chicago Tribune:

60 read the Chicago Tribune
40 read the New York Times
15 read the Wall Street Journal
25 read the Chicago Tribune and New York Times
8 read the New York Times and Wall Street Journal
3 read the Chicago Tribune and Wall Street Journal
1 read all three

a) How many read none of these papers?
b) How many read only the Chicago Tribune?
c) How many read neither the Chicago Tribune nor the New York Times?

2) A survey of families with 2 children is made, and the gender of the children is recorded. Describe the sample space and draw a tree diagram of this experiment.

3) A jar contains 3 white marbles, 2 yellow marbles, 4 red marbles, and 5 blue marbles. Two marbles are picked at random. What is the probability that
a) Both are blue?
b) Exactly 1 is blue?
c) At least 1 is blue?

4) A better coin is such that the probability of heads (H) is ¼ and the probability of tails (T) is 3/4. Show that in flipping this coin twice the events E and F defined below are independent.
E: A head turns up in the first throw
F: A tail turns up in the second throw

5) The records of Midwestern University show that in one semester, 38% of the students failed mathematics, 27% of the students failed physics, and 9% of the students failed mathematics and physics. A student is selected at random.

a) If a student failed physics, what is the probability that he or she failed mathematics?
b) If a student failed mathematics, what is the probability that he or she failed physics?
c) What is probability that he or she failed mathematics or physics?

6) In a certain population of people, 25% are blue-eyed and 75% are brown-eyed. Also, 10% of the blue eyed people are left-handed and 5% of the brown-eyed people are left-handed.

a) What is the probability that a person chosen at random is blue-eyed and left-handed?
b) What is the probability that a person chosen at random is left-handed?
c) What is the probability that a person is blue-eyed, given that the person is left-handed?

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1) In a survey of 125 college students, it was found that of three newspapers, the Wall Street Journal, New York Times, and Chicago Tribune:

60 read the Chicago Tribune, Denote by A
40 read the New York Times, Denote by B
15 read the Wall Street Journal, Denote by C
25 read the Chicago Tribune and New York Times , Denote by AB
8 read the New York Times and Wall Street Journal, Denote by BC
3 read the Chicago Tribune and Wall Street Journal, Denote by AC
1 read all three ,Denote by ABC

a) How many read none of these papers?
Solution. See Venn graph above, we can get
So,
So, 45 read none of these papers.

b) How many read only the Chicago Tribune?
Solution.

c) How many read neither the Chicago Tribune nor the New York Times?
Solution. See Venn graph above, we have

So,
So 50 read neither the Chicago Tribune nor the New York Times

2) A survey of families with 2 children is made, and the gender of the children is recorded. Describe the sample space and draw a tree diagram of this experiment.
Solution. Assume that A survey of families with 2 children is made. Then the sample space S is ...

Solution Summary

Six probability problems are solved using Venn diagrams, tree diagrams and proabability expressions.

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