MA 171 Week 5 Discussion

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Unit 5: Discussion

Remark: Each sub-question of questions 3 and 4 are considered to be different. Thus, 3(a) is a different question from 3(b). A student only needs to answer one sub-question from questions 3 and 4

1.      What exactly the Bayes theorem describes?

2.      A couple has two children and the older child is a boy. If the probabilities of having a boy or a girl are both equal , what is the probability that the couple has two boys?

3.      There are 2 urns. Urn A contains 2 white and 7 red balls, and Urn B contains 2 white and 5 red balls. Answer only one question.

1.      If a white ball is drawn, what is the probability that it came from Urn A?

2.      If a white ball is drawn, what is the probability that it came from Urn B?

3.      If a red ball is drawn, what is the probability that it came from Urn A?

4.      If a red ball is drawn, what is the probability that it came from Urn B?

4.      Suppose we have an urn that contains 4 white and 9 red balls. Two balls are drawn in succession without replacement. Answer only one question.

1.      If the second ball is white, what is the probability that the first ball was white?

2.      If the second ball is white, what is the probability that the first ball was red?

3.      If the second ball is red, what is the probability that the first ball was white?

4.      If the second ball is red, what is the probability that the first ball was red?

5.      Approximately 1% of women aged 40-50 have breast cancer. A woman with breast cancer has a 85% chance of a positive test from a mammogram, while a woman without has a 15% chance of a false positive result. What is the probability a woman has breast cancer given that she just had a positive test? 

6.      Suppose that an event A has a probability 0.4. Suppose moreover that P(B|A) = 0.3 and P(B|A') = .6. In the next two problems compute the given probabilities.

1.      P(A' ∩ B).

2.      P(A∩B).

7.      Give an example of random variable and an example of a non-random variable.

8.      What is the difference between an expected value and a mean?

9.      Describe when a given distribution is a probability distribution.

10. You draw and keep a single coin from from a bowl that contains 12 pennies, 9 dimes, and 20 quarters.
What is the expected value to you?

11. Repeat the previous problem but now with 5 pennies, 5 dimes, and 5 quarters.

12. A random variable X has range x₁, x₂, and x₃ all with equal probabilities. What is E(X)?

13. A random variable X has range x₁, ...,x_n all with equal probabilities. What is E(X)?

14. What is the expected value when we roll a die?

15. The random variable X has possible values X = a and X = -b with corresponding probabilities

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1.      What is E(X)?

2.      Suppose that you are playing a game with a fake coin that has a probability of Heads to be 1/3. You win $3 if you get a Head and loose $1 if you get a Tail. What would be the expected value? Would you say that you have more chance of a winning than loosing money?

3.      A box with 10 flashbulbs contains 3 defective ones. A random sample of two is selected and tested. Let X be the random variable associated with the number of defective bulbs. Find the probability distribution of X.

4.      Find the expected value of X the following probability distribution.