Six Probability Problems
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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
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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Solution Summary
Six probability problems are solved using Venn diagrams, tree diagrams and proabability expressions.
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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 ...
Education
- BSc , Wuhan Univ. China
- MA, Shandong Univ.
Recent Feedback
- "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
- "excellent work"
- "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
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